Illegibility: obviously, if really unreadable, it can't be delivered, and it's returned to the sender (when possible). All letters have their addresses scanned to turn the postcode into a computer readable form. Printed ones and some handwritten can be done all automatically. Those that fail are done by hand; someone looks at the address, reads it and interprets the postcode for the computer format. So if a person can't read it, they perhaps ask the person next to them to have a go, if no one can read it, it'll go back to the sender. This will get the letter to the right distribution office. On the actual round, it comes down to the postie to read it; again, if he can't it'll go back to the sender (when possible).
Without going into details, yes, the overwhelming majority of mail is checked for valid postage, mostly by machine.