Look it up, or talk to a person who speaks Parisian French. “Mais” does NOT sound like … I can’t do the accent with my keyboard. It does sound like the phoneme represented above.
I grant that there are different accents, but that’s not a tolerable mistake.
Yes, it is.
As with all rules, there are some exceptions to this one. Most notably, the schwa (/ə/) is permitted in word-final position (as in “paramecia”). Some odd interjections and colloquialisms may also get away with it, such as your “meh” (/mɛ/) and also “yeah” (/jæː/). But in general, you’re not going to find any words in Standard American English ending in /ɪ/, /ɛ/, /æ/, /ʊ/, or /ʌ/ (the lax vowels in the words “pit”, “pet”, “pat”, “put”, and “putt”). (YMMV if your dialect is sufficiently different.)
This causes problems for native English speakers not only with French but also for other languages which permit these vowels at the end of a word. For example, Icelandic is full of words ending in /ɪ/, and English speakers routinely pronounce them as ending in /i/. This can lead to confusion as there are some pairs of words which differ only in this final vowel.
To all the excellent explanations above, I want to add a piece of advice, as a fellow non-native speaker of French.
The one, single thing that you and your son can do for your pronunciation to sound more French and less American is to lose the [eɪ], especially in word-final position, and substitute the pure vowels [e] (for example, illimité) or [ɛ] (for example, mais).
It is not so important to distinguish between these two sounds, but it is important not to render them as an English-style [ei].