Привет!
Меня зовут Катя,
мне 23 и я мама
очаровательной
маленькой дочки.
Всего самого хорошего
и привет из Сибири!
Катя

Hi, my name is Katya, I'm 23 and a mother of a charming little girl. (literally 'daughter') All the best, and greetings from Siberia. Katya

Also, the н in Меня and мне look like they could be и, is that usual or just how this person writes it?
It's just a sloppy handwriting, don't try to copy that.
I've also noticed that many people who send me cards from Russia write т as a latin t and not like a latin m. Is this more common with the younger generation?
Yes, it's rather common among people of all ages. When kids learn to write they usually have a standard, nearly perfect handwriting. When a person grows older he or she develops his/her own handwriting, that suites their character and motor coordination. Many people include variations of block letters in their cursive. I personally often use "blok" Р's, H's, Г's and T's as capital letters, because they are faster to write.

In other words it's ok to write "т" like that, but note that such pseudo-block letters usually retain some features of cursive letters, for example "т"'s here are written in one stroke and are connected to other letters. If you just put a perfect block letter in the middle of a perfect cursive word it would stick out, but people will understand it anyway.