The number of credit inquiries on your credit history does have an impact (how big or small is debatable) on your credit rating. The assumption is that someone who solicits lots of credit lines may be overextending themselves. If one is shopping for a mortgage, however, it doesn't matter how many inquiries there are as it is expected to get several rates before choosing a mortgage.

I don't think you need to worry about 2 applications. In fact your credit score improves with higher credit limits (not used limits though) and a few (not many) lines of credits.

If your credit card is truly a no-fee credit card, there shouldn't be any fees posted to your account. Read your agreement terms - they'd list any fees if there were any.

Requesting a credit increase to an already existing account will not negatively affect your credit score.