Yes, the vowel "a" has the same pronunciation in both words.. The International Phonetic Alphabet (IPA) shows that each "a" has the sound "æ". hæd = had hæt = hat

Yes, it's the same vowel sound. To rhyme with "had" you could use "bad", "mad" or "sad." To rhyme with "hat" you could use "bat", "cat", "sat", "that", "rat", etc.

Both words have the same phoneme /æ/, but the vowel before /t/ (and any other "voiceless" consonant) will be slightly shortened (clipped) in native English speech. In NYC accent, /æ/ is usually /eᵊ/ or /ɛᵊ/ in one syllable words ending in /d/ (and some other sounds), but this is *not* the case with “had”, the sound change does not apply to past-tense verb forms, unstressed modal auxiliaries, and some function words. I can go into more detail if are you really interested. Note that "hat" and "had" do not sound the same to any native English speakers, nor do they rhyme.

The vowels rhyme, Alyosha, and it is indeed the same: /æ/. But note that whereas HAT usually has it's full pronunciation /hæt/, the word HAD as part of a verbal phrase is often unstressed and short, closer to /hɛd/: He had called me [ˌhihɛd ˈkʰɑːɫd.mi]. And it can be made even shorter than that, in which case it's just contracted to the word in front of it: He'd called me. [ˌhiːd ˈkʰɑːɫd.mi].