Getting paid

Does it look like the hospital really needs you for staffing? Then call HR and say you will not work the next shift until you get a deposit in your bank account. In the 80s (I am old) I was the first PRN nurse at my local hospital. I negotiated my pay rate etc...except they never deposited my pay into my bank account for hours worked. Finally I waited until a weekend and said "If you don't pay me I am not willing to work this weekend". When I stopped at HR before my shift,  my check was waiting for me. 

The agency is your employer of record and should be trying much harder. I can think of a number of ways to problem solve here. The easiest would be to simply fill out a simple timesheet documenting all your hours worked (old fashioned but still common for travelers) and get your director/manager/charge to sign it and you can email/fax to your agency. A second way would be to get one of the above to print out a copy of your hours that have been entered into the third party software. Believe me, the hospital can do this, and are indeed responsible for reporting your hours to your agency.

While you are on this mission, ask for a log in to the pay reporting system. All employees (and travelers) should have an account so they can check that their time is correct. Hard to imagine any Kronos competitor not providing this. Then you can print it out yourself and send it to your agency. I've had to do this myself with incompetent agencies and hospitals.

Scrubs? That is pretty trivial. Check your contract or handbook. I would assume there is a schedule for when they reimburse you. Perhaps after 3 weeks, or even at the end of the assignment.