I would first try a silicone spray inside the lock, use sparingly. Then remove the small triangular piece of plastic in the top of the door jam (creates access to the rear of the lock), and spray a little white lithium grease on the back of the lock and rod mechanism. Silicone can be used there also, but the grease lasts longer.

As for the remote, that is a gamble because you do not have a model number. Get one from Ebay or where ever, then turn the key to position II 5 times and leave it in position II. You will note that the red led on the dash will blink rapidly.

With the door closed, push the lock/unlock button, and wait for 10 seconds or till the led goes out. Then the remote should work.

Do a search to find the correct method, my memory isn't what it used to be.