*Yes, you can be made a princess in your own right rather than only being born one. Incidentally, though Princess Marina didn't have to give up her own title (presumably because a fourth son was pretty far down the line), Prince Philip had to so when he took British citizenship and got engaged, he was Lt Philip Mountbatten RN, but was made Royal Duke on the morning of his wedding day; he wasn't actually a prince again till the Queen made him one again in the 50s. There was a bit of a stushie when George was born as William described Catherine on the birth certificate as Princess of the UK when she wasn't made a princess. I don't know the accuracy of all that though.
Princess Michael of Kent is only a princess by using her husband's name and not in reality. I think the Kents get a bit of slack due to losing their father so young - wasn't Michael a tiny baby at the time? Otherwise I bet the Queen would have gone NC with the Michaels just as she did with the Windsors until he was dying.*
Alice wasn't created a princess in her own right, she was just allowed to style herself as Princess Alice, Duchess of Gloucester. There was no LP's issued or anything. She was also a princess by marriage, but HRH Princess Henry of Gloucester isn't much better than HRH The Dowager Duchess of Gloucester really.
The fuss over George's birth certificate was a nonsense really. The wife of a Prince in the UK is a Princess, regardless of what title she actually uses. It's like the whole thing about Camilla not being Queen when Charles is King - no matter what they call her she will be his Queen consort, just as she is HRH The Princess of Wales at the moment.
Michael was only a month or so old when his father died. The Kents, and the Duke of Gloucester, did a lot of work on behalf of the Queen when they were all younger. Not Michael as much, but the Dukes and Princess Alexandra did a lot of engagements before the Queen's children were old enough to do them so that also gets them a lot of slack from the Queen.