I fly every year in November to Las Vegas to visit friends and use Lufthansa to either Los Angeles or (in the past , now no more Phoenix) . Connecting with South West onward was no problem if one left 2 or so hours between flights . By the way the family cleared Immigration at LAX last month in 20 mins !!! Booking online from Europe with a European credit card is no problem and you just turn up with a PNR .