Hello!
Yes, you could say "Tyttö aikoo tulla insinööriksi", although it would sound kind of clumsy even though there's nothing actually wrong with it. "Aikoa" expresses the same on its own, so it would be better to say "Tyttö aikoo insinööriksi". :)
It's really hard to give an exact answer on when to use "tulla" and when not. It depends so much on what you are trying to say... Hmm... I'd say "tulla" is usually used when it is NOT the only verb in a sentence, though even then it is mostly optional. Usually it doesn't change the meaning that much at all. The translative ending expresses te verb "to become" as well, so in Finnish it's up to you if you want to use "tulla" with another verb already in the sentence, or not. Here, I'll show you:
Mikko haluaa palomieheksi. (Mikko wants to become a firefighter)
Mikko aikoo palomieheksi. (Mikko wants/aims to become a firefighter)
Mikko haluaa tulla palomieheksi. (Mikko wants to become a firefighter)
I'd say you shouldn't worry that much about the verbs you use with translative. Just remember that when something becomes or transforms/turns into something, use translative endings!
Hän haluaa/aikoo opettajaksi. (He/she wants to become a teacher)
Hän muuttui vihreäksi kateudesta. (He/she turned green of envy)
Huomenna sää muuttuu sateiseksi. (Tomorrow the weather will turn rainy)
Do you know something about Pokemon? Some people liked my Pokemon example; "Remember that Pokemon evolve in translative" --> Pikachu kehittyi/muuttui RaichuKSI. ;)
About the second question:
We're talking about nuances here, although the second might sound a bit odd sometimes. There isn't really that that big a difference between the two phrases, but use the first one, it will always be correct when you give presents, although the two phrases both mean the same thing in English. :) The first just sounds more natural!
I'm sorry about the long answer... I hope I helped you, even a little!