not familiar with the industry practice nowadays

but in my last life when I had to fly for interviews, flight tickets, airpot pickup and dropoff, and hotels were all pre arranged and paid by the companies that conduct interviews.. its a little risky to pay out of your own pocket and expect to receive reimbursement in an interview situation..
I'd suggect to send an email to HR in a very professional and firm tone demanding a speedy resolution by a certain date, otherwise, you'll pursue a legal process against them after the dealine. Maybe you should also post this in the legal forum next door for better advice?