I m curious also with changing your battery chifx45, but please post your results like Andrew mentioned.
Im just skeptical because once your car starts, doesnt your alternator kick in to supply the proper amp/volt?!
im thinking it hesitates possibly due to you flooding the engine during the starts.
if your air filter is that dirty, I agree with others that it is a dirty TB.
my FX was doing the same thing 2 years ago. i thought the same thing - maybe a tune-up was needed or the intake was dirty or the filter because of CAI. i was planning on doing some maintenance until i got in one day after work and it was dead lol. i jumped it and went and swapped the battery and it eliminated the hard start and rough initial idle. doing the same thing again now and almost didn't turn over this morning.
i'll post results once i swap the battery out again, but, that's why i was saying to swap the battery now. it isn't difficult to get to the TB, etc. but i don't believe it's necessary to even check it.