H1 B and expenses

[helppls]

if you are working for a consulting company on W2 and working at a client site. Can you submitt the bills directly to your client for the expenses incured at client site and get the check from the clent directly on your name?

On H1 can you accept checks from anyone else other than your employer?

Please advice me as I have lotof money struck as expenses and I don't know which way to get them back. Please help......................