If the pump still works, then you didn't damage anything by failing to remove the screws.
Is it just the hot air gun not working? I do not know about the errors, but it is possible the element or the temperature sensor in the handle has gone open circuit. They are cheap, and if it is the case with yours, the seller should be happy to send you a replacement. You can open up the handle and check the resistance between the leads. The units do not have micro's from memory - just some Intersil chips to do the A/D and drive the display and so you could find the datasheet and find out the meaning of the error messages. I am surprised they are putting up non-numeric codes, and there is a possibility that a display circuit is faulty. Normally, the display will either show a number, or if the voltage to the Intersil chip is too high, it shows a "1" followed by 3 blanked out digits. The "5-E" just does not sound right assuming your unit does have a Intersil chip.
You will want to get some spare heater and iron elements anyway. They are on ebay. If you cannot find them for your model, search for "Kada 852D+" on ebay. Get some decent Hakko tips as well.
Edit: Just read you have solved the problem. Excellent.