Yes you can.
However it all depends on your wife who must exercise one of her EU "Treaty Rights2 to remain in the UK. It could be one of the followings:
-Worker(employment including self-employed);
-Seeking employment;
-Self sufficiency (this is the case for pensioner for instance);
-Study.

‘Self sufficiency’ means having necessary saving, receiving regular funds from abroad... and can also include where it is the non-European national who is working and supporting the family, rather than the European national themselves.

In a nutshell they want to avoid the EEA to claim benefits and public funding, so if this is not the case, it is usually pretty easy to demonstrate that a Treaty Right is being exercised.

You will find all the guidance regarding EEA on the UK Border Agency website.

And more details also on the UK work permits website.

 

You will need to re-apply for another EEA family member visa.

 

Yes, I responded.

 

In theory, as a EEA, she has right to stay (with her family) as long as she exercise one of her European treaty right, which include looking for a job (in that case being of the process of getting licence).
Any proof (including studies/exams) that she is ercising treaty rights are good.In term of your family visa, it should have a date on it. You will have to renew it before it expire.