(1) Correct: No French consonants are normally aspirated. In English, not only [t] and [k] are aspirated, but also [p]. This is the natural class of voiceless stops. They are usually aspirated at the beginning of a word or stressed syllable except in certain clusters. Because this behaviour is allophonic and therefore unconscious, it's hard for English speakers to unlearn it in French.
Note that if you were speaking French, the word aspiration often refers to another phenomenon (an h aspiré is one that blocks elision and liaison). It can also have the meaning above, but the short Wikipedia article gives you an idea of how little the English meaning is needed in French phonetics.
(2) I'm not sure exactly what you mean, but I'll try to answer your question.
(a) By "bilabial" I suppose you mean "with rounded lips"? French certainly does have more rounded vowels than English: [y], [œ], [ø] for example. One reason these vowels are particularly salient for English speakers is that they are all front vowels, and we do not have rounding on any front vowel.
However, it's clear that French has plenty of unrounded vowels. If you compare [i] and [y] (lit / lu), [e] and [ø] (des / deux), [ɛ] and [œ] (sel / seul) you'll find that the first in each pair is not rounded.
[ɛ̃] is an example of a French nasal vowel that is not rounded.
(b) A vowel never has the tongue touching the alveolar ridge; that would be an obstruent. If you mean a front vowel, French has those, as we've just seen. But there are also vowels pronounced midway in the mouth (schwa) and vowels pronounced at the back (e.g. [o], [u]), like any language.
[ɔ̃] is an example of a French nasal vowel that is not articulated at the front of the mouth.
However, all French back vowels are rounded. Therefore, every French nasal vowel is either front or rounded. But they don't have to be both.
(3) (I read your last paragraph as a different question.)
Nasal vowels behave differently depending on the language. In English, nasal vowels are often the result of contact with other nasal segments. They are not contrastive; that is, they don't change the meaning. They are allophonic.
But owing to the history of French, nasal vowels are often left where a nasal consonant once was. Compare sais /sɛ/ vs. sain /sɛ̃/. Now, despite the spelling, neither word phonologically has an /n/ in the present day, so we can't attribute the nasality to assimilation or context. The story involves the history of French.
During this history, these syllable-final consonants disappeared and left the nasality on the vowel as their only trace. In fact, when the nasal consonant is still pronounced, the vowel is not nasal, oddly enough!
For example, in the word don the "n" is not pronounced, so the /ɔ/ is nasalized: /dɔ̃/
But in the word donne the "n" is heard, so the /ɔ/ is not nasalized: /dɔn/
A more thorough examination of how this happened is not fitting here. But you can read about some interesting exceptions here and here. Also see the end of Circeus's answer.