No, the A in phat is the same A as in “father.” Definitely not the schwa sound in “putt,” as Thai does not have that sound. Phat rhymes with “pot,” “rot” etc.
I was wrong, though, in that ผัด ends with one of the T letters. That last letter IS the D letter, but it’s pronounced as D only at the beginning of syllables, similar to how ล is pronounced as an L at the beginning of syllables but as an N at the end. The T/D difference is not as stark, however, and so it is often still transliterated as a D.
Siam, it really depends on the way you speak English. I was going to use “putt” as an example too - I don’t use the schwa sound in putt, for me it has almost the vowel sound as “father”, just shorter (I think). ผัด certainly does not rhyme with “pot” or “rot” the way I pronounce those words.
And I thought you were correct (if perhaps potentially misleading) when you described ด as a T letter, because like you said, it acts exactly like one at the end of a syllable, just like ฃ, จ, ต, etc. do.
Yes, I guess there will be differences in how different native English speakers say those words. For my particular accent, “pot” sounds exactly like the Thai phat (never mind the tone). Perhaps its easier to say the A in phat is like saying “Ahhhh” for the doctor, only VERY short.