Are there places to safely save overflow inventory/stuff? I saw I can expand "bank" storage from Crazy Earl, but where is the bank, and what can I store there? How much? Are there multiple locations?
The bank is in the crimson headquarters in Sanctuary. I cannot confirm if there are multiple locations.
To gain access to it, do the main story line quests - you can't miss it. (you get access to the bank from Roland)
You can store anything you can hold in your inventory in your bank. The storage of the bank can also be upgraded with storage deck upgrades from Crazy Earl.
The claptrap also gives you a side quest in Sanctuary to obtain gratuitous amounts rocks (138,000+), and a bunch of other random [ and hilarious ] items for the reward of his secret stash! -- the ability to store items in a locker which you can pass those items between character ( for twinking, etc)
In True Vault Hunter Mode, you can access the Bank near the start of the game at Claptrap's Place, Windshear Waste.
As commented by @Hammar, also in True Vault Hunter Mode, you can access Claptrap's Secret Stash behind the door just off the right edge of the above screenshot (in a closet where several broken claptraps are stored and the first "Cult of the Vault" symbol can be found).