This all depends on the individual appraiser to whom the appraisal assignment is given to. With appraisal management companies getting most of the work, they generally push for a 2-3 working day turn-around. However, it should not take longer than a week for a single family home.
Update: I just read your question again and notice that I have not answered it completely. But here we go. If you already waited 2 weeks since the appraisal inspection that's an awful long time in today's market environment. You have the right to contact the bank or lender and inquire about the status of the loan. Tell them the appraiser told you it should be ready in about a week.
No, the appraiser will not contact you to fix any deficiencies noted. They will be listed on the appraisal and are known as conditions to the appraised value. It is then up to the bank underwriter to waive these appraisal conditions or ask the seller to fix the deficiencies before the loan will be funded. In rare cases the amount to fix can be withheld in escrow and the loan can close, but I don't know if HUD rules allow this for FHA loans. Good luck.
Real Estate Appraiser