You don't make a deposit at the book. you basically "buy" a ticket for whatever wagers you want to make at the time. You don't walk in and open an account or anything... its just a counter with a big board behind it with all the lines. You walk up and say "$100 on the yankees to win the pennant", hand them a c-note, they take the money and hand you a printed ticket with your bet and payout and a barcode on it.
There's quite a bit of room between casinos on the strip. Depends what you consider the majors. here's a map:
http://www.lasvegasmaps.com/stripmap.html
Tropicana to Flamingo (Caesers to MGM) is a bit over a mile. Probably a mile and a quarter door to door.
The green tube on there is the monorail. buy a 10-trip pass or a 24-hour all you can ride pass if you're really going to be hopping a lot.
Its vegas. the bank isn't going to blink at you walking in and wanting to withdraw five figures, assuming the money is free and clear in your account. Citibank has a huge presence in Vegas, just locate a branch and take a cab.