An FAA *and* an ICAO license? Since the FAA certificates are ICAO compliant and ICAO doesn't issue licenses, I am assuming you mean you have a license from another country that is also ICAO compliant.
In answer to your first question, so long as you take the practical test for the ATP within 24 months of passing the written, there are no other restrictions.
As to your other question about the type ratings: legally according to 14 CFR 61.41, your ICAO compliant training is acceptable and your instructor can give you the required training endorsement. However, I highly doubt your local DPE is going to be familiar with and comfortable with the process. Best bet would be to visit the FSDO to see about adding the SIC types on the basis of your foreign training.
Last edited by raysalmon; 25th Jul 2014 at 15:15.