A good place for you to start is with the FAQ on UK EEA Family Permit.

For unmarried partners

Another helpful article is Obtaining a Visa to Live, Study and Work: the United Kingdom.

The official site offers more info: http://www.ukba.homeoffice.gov.uk/visas ... s-settled/

Are you on a visitor visa? Then you may need to return to Israel to file: http://www.ukba.homeoffice.gov.uk/visas ... k/provides more info on settling with partners, and http://www.ukba.homeoffice.gov.uk/count ... %20English provides info on applying from Israel.

You must show that:
you and your partner are both aged 18 or over at the date of application;
your partner is not related to you in a way that means you could not marry in UK law;
you and your partner have met in person;
your relationship with your partner is genuine and subsisting;
if you are married or in a civil partnership, your marriage or civil partnership is valid in UK law;
you meet the suitability requirements;
any previous relationship has permanently broken down (this does not apply to certain polygamous relationships);
you and your partner intend to live together permanently in the UK;
you meet the financial requirement;
you meet the English language requirement; and
if you are in the UK and want to extend your leave or apply for settlement in the UK you will need to meet the suitability requirement.

I also recommend you search the forums as they are a great resource of other people's experience. Best of luck & let us know what other questions you have.

 

As noted above, You cannot apply to extend your leave to remain if you are in the UK as a visitor. So yes, you will probably need to go back to Israel. Start on the UK Border site here: http://www.ukba.homeoffice.gov.uk/visas ... you-apply/

Yes, if you married might help. Here is info on the UK Marriage Visa. I would also recommend searching the forum as many people have shared info about the documents they submitted to provide proof of their relationship. Collect all the tangible evidence you have: joint bills, valentines day gifts/cards, letters/postcards, etc.

 

I ma not an immigration law expert but I am strongly agreed the this that you cannot apply to extend your leave to remain if you are in the UK as a visitor.
Yoy are bound by immigration law of the country.