You'll need an M visa and go through the TSA process.
All hardly possible in one week.
There are no type ratings below 12500lbs, so no, the 300 will not appear on your FAA CERTIFICATE.
There are no LICENSES in the USA.
Insurance is included in the rental rate.
An Austrian prof check is not sufficient. You have to do a Biannual Flight Review (BFR), read FAR 61.56.