The 420a engine is just not built for boost. You need to upgrade internals in order to even get past 5 psi. The 4g63 was built for boost and can handle alot of power on stock internals. You could probably find a turbo dsm for the price of an NT which is why it is easier to just get a 4g63.
Welcome to the site. You can find almost any information on the site just by searching but dont be afraid to ask if you need help. Good luck with the projects and get some pics up.
Try disconnecting the battery for awhile then hook it back up and see if it works. This used to work on my old NT which could be an indication of tcu(transmission computer) It would be much easier if it was just a tcu problem and not the actual transmission you could just get a used one if it is.
If you dont want to go to the junkyard, you could check miller import parts. Its a bit more expensive but you wont have to make the trip or get dirty lol.
I would change that air filter to a k & n. Fix those oil leaks and then boost leak test to get it running the best you can.it doesn't look like you have too many mods there. You have a greddy bov recirculated which is good and what looks like a 14b. It wouldn't hurt to clean up that engine bay...
Try wrapping it with exhaust wrap, it should help with the heat. You probably need to get a cat back for it to sound any better. You should also make sure you dont have any exhaust leaks.
you are suppose to have the stock airbox cover the k&n filter. Most places here in CA dont really care about that but you get some that follow everything by the book.
I have heard of people passing emissions with no cat but it would probably be easier to pass with one and maybe just get a 1g bov so you can recirculate it. Also wouldnt having no cat cause a cel and cause you to fail emissions?
My first car was a 420a eclipse and i loved it. IMO it just not worth dumping the money into it. If i could go back in time i would have just done all the maintenance and kept it looking nice. Later you can step up to a 4g63t.
Not sure what you are looking at from there but I would suggest buying from a supporting vendor like extremepsi.com. You will never have to worry about them being legit. Sorry if I didnt answer your question.