Most American speakers use more-or-less the vowel of RP (Received Pronunciation, the most common or standard "reference" British English accent) "grass", "fast" and "arm" in all of pasta, father, mop, don. But not lost; that has a different vowel in "GA" ("General American," the "standard" reference American accent). "Lost" and "mop" have different vowels in "General American" English due to a vowel change similar to the one that is responsible for the different vowels of "last" and "lap" in RP British English. "Pasta" and "father" have the same vowel as "mop," but not the same vowel as "lost" in GA.
The vowel in the word "father" is typically written /ɑː/ when transcribing British English, with a vowl length marker (ː) because British English is often analyzed as having phonological vowel length.
Vowel length is less important (or at least, less obvious) in the phonological system of American English, so usually it is just written /ɑ/ when transcribing American speech.
Of course, as with all IPA vowel symbols, this is a simplified representation of a variable set of actual vowel sounds.
Distribution of /ɑ/ in a typical American English accent
You can see some explanation in the Wikipedia article Pronunciation of English ⟨a⟩. Basically, /ɑ/ is usual in rhotic American English accents for father, before /r/ (also analyzed as a unitary rhotic vowel /ɑ˞/), and for the majority of speakers, in some words that historically had a "short o" such as lot. The exception is words like cloth and lost where "short o" ended up being changed to the "aw" vowel of thought.
Words like palm also historically had /ɑ/, but the common restoration of /l/ has caused some speakers to change the vowel to the thought vowel (/ɔ/).
Both interdental fricatives found in English (/ð/ as in "father", and /θ/ as in "thank") are uncommon among other human languages, consequently many non-native English speakers (and speakers of some native dialects, like Cajun) can have a hard time learning to pronounce them. As mentioned in the comments (thanks @aparente001), the substitutions made by non-native speakers (like /d/ and /t/) can spread to native speakers through media and pop culture. This may be especially common in song because of the demands to rapidly articulate consonants between notes.
However, you are right to study and practice as you are, because if in everyday speech, you fail to pronounce the interdental fricatives where appropriate, your performance will strike native speakers as foreign.
Best Answer
During the Irish potato famine 1845-1852 more than a million Irish emigrated to America.
This influx probably affected pronunciation in the areas of America where immigrants were concentrated.