I have requested a copy of my contract but 10 days later still haven't received it. How long before they have to / should respond?
I would NOT work for a company which does not provide a valid, properly-signed-by-all-parties contract, LATEST during the first day of work.
I would make an exception (but not more than a few days) ONLY if they provide a very reasonable reason.
If there is a reasonable motive to not have a proper contract in due time (i.e., more than a few days), then the company should release you a kind of "pre-contract", where they specify all the conditions of the contract, AND they mention that the proper contract will follow latest after XYZ days.
"How long before they have to / should respond?"
In most of the cases, for a contract being finalized, every involved party gets a valid copy of the contract, with all the authorization (signature, seal etc as applicable) as the contract gets signed. Considering the time taken for doing the due diligence (taking print out etc), I see no reason for not getting a copy on the same day as it's finalized / signed.
Unless you have a copy (either in an electronic format or a hard-copy/print out), there's no way you can claim anything about the contract.
Follow up immediately and push for getting yourself a copy ASAP.
If you accept the offer, the contract may take a very long time to arrive.
It may actually never arrive.
If you say, let me take a look at the contract before I accept, the contract is FedExed to you the next day (or in the next two business days at the very most).
Note: Please replace the word FedEx with the fastest method of delivery in your area. It could be DHL, Express mail, a messenger on a bike. Whatever is the fastest method of delivery in your area (assuming they don't use email or electronic signatures).