I speak persian. Persian grammer is somehow near to hindi. There isn't articles in hindi or persian. So when we say "I eat carrot" a/an/the isn't needed. We mean the carrot itself, not it's quantity. In persian we use carrot in single form, not carrots. Unless they number or quantity of it is asked. So although एक गाच़र is right, it isn't necessary.
Yes, I also noticed this - the full sentence is pronounced as gaajar, but if you click on just the word to hear it again it's pronounced gaazar. I've noticed this outside of explicitly learning Hindi, e.g. in cooking/eating/watching films it's kabhi sabji kabhi sabzi.
It's my understanding that words with 'z' are generally older loan words, and some people (or regional accents) will pronounce them 'j', as if the lower-left dot (bindi?) weren't there, and sometimes omit it in writing too.
But if 'gaajar' here is the truest spelling, that doesn't explain why it would be pronounced as if a dot has been introduced.
EDIT: Ahah! It's a 'doublet' - it has two roots - from Sanskrit gaarjar and Persian gazar. By some combination of the two (very literally, since all three languages use Devanagari) Hindi gets gaajar, but I suppose people pronounce it in varying ways closer to Sanskrit or Persian according to region or other influence.
Fascinating observation man! I'm guessing this same thing would happen with other words, maybe even if they don't have a dual root. Or, maybe dual root isn't quite the right phrase. They evidently have some similar ancient root that has diverged somewhat, with Persian taking on a z quality, and Hindi taking a j. This little point of confusion has really become very interesting.
"vah" is the actual pronunciation of वह. Unlike the pronunciation of वहां which is "vahaan" with a nasal "n" after the long "a" To pronounce वह "vo" is rather colloquial. (https://www.duolingo.com/skill/hi/Basics-1/tips-and-notes)