Nice find! Looks like a great goat. Hard to find unmolested gto's now-a-days.
The shifter looks aftermarket with a different shift knob. The shift knob doesn't look familiar. Somebody installed some kind of after market short throw shifter and probably just bought a random shift knob they liked. Almost looks like a corvette shift knob.
The 1-4 skip skip shift will only happen if certain paramaters are met while driving. It's dependent on speed and rpm so it won't happen all the time. Or somebody may have put a skip shift eliminator on it which is a connector you plug into the trans under the car the overrides the skip shift function. Or someone had it tuned out if they had the car tuned or had a hand held programmer. It's an annoying feature really so be glad that you haven't had to deal with it lol.
Programming additional key fobs is a pain in the butt. If you really want to get more fobs you'll need to do a search and find the process.
If you're looking for parts in general for the gto the best sites are:
Welcome : MarylandSpeed.com- Your First Stop on the Way to Performance!
GTOG8TA.COM - Late Model Pontiac Performance and Restoration Parts - Home
For that specific lens or light fixture on the lower interior light panel, you better off checking ebay or looking at the ls1gto forums on the for sale section. People parts these cars out all the time and have little random things like that listed for sale. You're really not going to find little things like that new anymore since the car is over 10 years old and new factory parts stopped being made years ago.
If you're wanting to upgrade anything look into the suspension first. The factory springs sag bad causing uneven tire wear and the shocks/struts are undoubtedly worn out. It will most likely need Front Radius Rod bushings and Strut Mount & Bearings. You can get a whole package from Kollar Racing. Andy Kollar is the best guy on the planet to talk to when it comes to setting you up with the proper suspension pieces either for Daily driving, drag racing, or turning corners. You won't believe how nice these cars drive with updated suspension parts.
A couple of other things to check... Check that the fuel line on the backside of the motor is not rubbing on the fuel rail cover. It can slice a whole in it over time and cause fuel to leak out onto your hot engine. It's recommended that you check behind the glove box and make sure the wire bundle on the right side isn't rubbing thru on the metal bracket. There are untold amounts of people that have had their BCM short out from this. And there's no way to buy a new BCM anymore. Get under the car take a look at all the bushings and joints and make sure everything looks ok. Bushings can get super worn out causing problems. Also check the driveshaft carrier bearing. It's in the center of the driveshaft and holds the 2 peices of it together. The rubber ring that supports the drive shaft can rip and cause the DS to flop around and can wipe out your tailshaft bushing or pinion bushing.
It's also a good idea to swap the trans fluid and diff fluid as you don't know how long it's been since someone did that.
The trans takes 4.6 quarts of GM Genuine Crank Case Fluid part#88861800 https://www.amazon.com/Genuine-Fluid.../dp/B00BK7M1EK
The diff takes about a bottle and a half of Torco 85-140 RGO with 2 ounces of Type F Friction modifier. You'll want to get a bottle top pump that you can pump it into the diff in as well.
Type F firction modifier: https://www.amazon.com/Torco-AFM0050...V9FNR9YDPM2SAG
That's the basics. If you have any other questions fire away! And enjoy your new ride!