First, if your father is an Italian citizen and your mother is married to him, of course, she can apply for an EEA permit to accompany him to the UK, or to join him there. You did not say where she is residing at present. Of course, father would be supporting her application, and would have to deal with some of the questions on it that I think you are already familiar with.
As for yourself, you say you have a valid Italian residence permit, but exactly what type of permit is it? Do you mean that you have the right to live in Italy without restrictions, or is it just for study? You need to clarify to get a proper answer.

 

As I've been through the process myself, I can tell you that first, you would have to legally live in the UK for a minimum of 3 years without restriction on your stay, in order to pay the EU rate of study fees. That means you would need to have a UK residence certificate for at least 3 years, and time spent as a student does not count.
Unless you are qualified in one of the jobs on the UK s shortage list for highly skilled and skilled workers, you will not be allowed to work at all in the UK. Except, if you were a full time student at Batchelor level or above, at a UK university, you may be able to work part-for a maximum of 20 hours a week. There are now serious penalties for working in excess of that, which can now result in cancellation of visa and goodbye UK. In any case, part- time jobs are not easy to find these days, especially around university cities.
Regarding over age dependency, that could be open to you IF you qualify, but from what you've said, it appears you won't. The EU rules on dependency relating to children over 21 requires proof of dependency at the time of appiication. It does not mean totally financial dependency, but evidence of some financial dependency would be expected. Dependency for accommodation, and moral dependency can be taken into consideration too, but the state to which application is made has considerable discretion for refusal. In the UK historical dependency is scrutinised. In the case of MY dependent who has lived with me here in the UK since 2008, while studying, and who I fully supported financially since 2008 until mid 2013 when she got a part-time job, and who I still partly support financially, her application was rejected the first time because Immigration was not satisfied with the level of personal medical insurance she had.
That brings me to my last point. The UK requires non-EEA over age dependents to have COMPREHENSIVE private medical insurance, for the whole time of their residence until such times as they are granted permanent residence. The fact they might be covered by the UK's comprehensive National Health Service, or that of any other country, is not considered to meet the requirement. In practise, an over 21 DEPENDENT, can apply for a UK Family Permit to enter the UK, and if granted, can then apply for Residence from inside the UK. That insurance cover is mandatory to get the Residence. My daughter's cover comes from AVIVA and it IS comprehensive. It took a while to find a company that knew anything about satisfying Immigration requirements at a reasonable cost. At your age it should be OK. For me...totally unaffordable, but luckily, as an EU citizen I don't need it.
If you can give more specific details, I'm sure there will be someone on here who can give you more help than me. I can only go by the experiences of myself and my family members.
Talk to your father!
Good luck!

 

I'm afraid an economics graduate, even with a Masters or a PhD is not a discipline on the UK government shortage list, and indeed, many of our own graduates can't find employment in that field, because there are far too many of them. The total number of highly skilled immigrants is now limited to a certain number each year, and apart from that, non-EEA citizen applicants will have to find a position which guarantees a minimum salary set by the government, for each job on the shortage list, and some are very high. The list is supposed to attract the best. As an example, the only two engineering jobs on the list when I last checked it were 1) nuclear engineers wanted for decommissioning nuclear power stations, and 2) geo-technical engineers for the construction industry. The list is regularly updated. You should be able to find it if you 'Google' for it. Of course, if you can get to be in the UK as a dependent, you will be able to take any job you can find, but there is a sting in the tail. If you earn too much, you probably won't be considered a dependent, and the 'too much' is not very high, and is 'at the discretion ' of Immigration. I don't think there are too many normal healthy, age over 21 but under 40 'children' getting residence permits. Its not what was intended for that classification. Most will be those who are over 21 but who are unable or unfit to support themselves.

 

Immigration will be looking for evidence of regular money transfers to your bank account. Paying student fees is not considered 'essential', but paying for food, accommodation, clothing etc, is. Your father would have to first find a job that would enable him to support you. What might come from other relatives does not count when it comes to someone claiming they are supporting a dependent. You have to realise that the UK is constantly tring to find excuses for NOT letting immigrants in, so if Immigration finds the slightest thing wron with an application, they will reject it, as they did with us.
If you are admitted as a dependent and are then successful in getting a Residence Certificate, it seems you will be expected to remain a dependent for 5 years, before you can apply for permanent residence. That might change for the worse, because if you have been keeping up with the news, because of the last week's European Parliament election results, several countries, including the UK, France, Germany and Italy are demanding changes to EU rules, to hand back immigration control to the individual countries, and to severely restrict immigration, even between EU countries, unless the person has a firm job, and for them to go back to their own countries if they lose their job before they can become permanent residents. The UK will be holding a referendum on whether to leave the EU, and there is a lot of support for leaving, but not from me! We can't guess the results, but if the UK leaves, it won't be the only one, and it will quickly block immigration from EU countries. It has already announced the UK will puts a new Human Rights Law of its own in place of the EU one. Times are changing, and not for the better. Italy is also demanding big changes to human rights laws so it can refuse many of those claiming asylum, and kick out those being smuggled from North Africa.

 

I don't know what you mean regarding a 'stay permit (a kind of green card)' because there is no such thing in the EU. The under age 21 child of an EU citizen does not need a permit to stay in Italy. What I don't understand, is if you were living in Italy before you became 21,why did your father not apply for your citizenship? Maybe you should try to work in Italy until you can apply for citizenship in your own right. If you are able to gain residence in the UK, you will have to wait for at least 5 years before you could apply for permanent residence, and that could be lost if you left the UK for a period of more than 2 years. You have some tough choices to consider. I hope it goes well for you.

 

If your father got citizenship when you were 19, if you were in Italy at the time, you will already have 4 years towards your residence requirement, won't you? I assume before your father got citizenship he already had a residence permit for several years, so if you were in Italy earlier, that can also be used to help you. It seems one way or another, you could get your Italian citizenship soon. If you can survive until then, you will be able to work anywhere you like in the EU.
Italy was one of the original six founder member countries of the European Union (EU) As far as immigration law is concerned, Italy must strictly comply with European Law which is the same for all member states. Italy can't have laws which conflict with EU law. Italy is allowed to give BETTER terms than the EU requires, but not worse, otherwise it would be penalised by fines from the EU and the European Commission would order it to change.
If you ask questions about EU immigration law, it would be helpful if you give the fullest details about yourself and your circumstances, but of course, not your identity. For example, in addition to your age and present immigration status. you need to state exactly how long you have been living in Italy, and who with. If with different persons, when and how long with each. Also, where you lived before, and with whom. You don't have to be living in the home of a sponsor to be dependent on them, but if you do, it helps. If you are thinking of applying for an UK EEA Family Permit it would greatly help if you could show there was a pattern of regular financial help from your father. In other words, ask him to transfer some money to you every month starting as soon as possible. The law does not say you have to keep it. You could just draw it out in cash, and hand it straight back. It is the record that counts. It is not illegal. If he is supporting you, I'm sure your father must be paying bills on your behalf. May be tuition fees as well. Even plane tickets. If you can put together that type of thing it all goes towards proving dependency. Perhaps you can discuss it with your father.
One other thing, about work opportunities. They are not very good here either. My daughter has a Master of Engineering from a top British university, where she came near the top of her class, but she has not found employment as an engineer. She is still looking but is working as a self employed interpreter/ translator.
However, I would think that for you, anywhere in the EU would be better than going back to Bangladesh. That country's economy is falling apart, and there seems little chance of it getting any better in the foreseeable future.
Anyway, don't give up your hopes. There is an opportunity for you somewhere.
Good luck