Inland GI M1 Carbine question

[NRA]

'does not have matching serial numbers'

No carbine originally had matching numbers. The only thing numbered is the receiver. Carbines with numbered parts were previously owned by Bavarian police and refinished/numbered there before being sold back into the USA by an importer. Your carbine appears to have its rebuild refinish in very good condition and most likely contains a mixture of parts. Looks like a Winchester stock, for example.
When you say 'Inland', are you going by the barrel? You go by what's on the receiver under the rear sight overhang, but manufacturer can also be told by the serial number.
I don't know of any competition using the carbine. They are pretty accurate anyway if everything's up to spec.
The bolt will need to be disassembled, cleaned, inspected and put back together. A short-cut is to blast it out with carb spray, working the extractor and ejector, then soak in Marvel Mystery Oil. Drain off excess and check the extractor and ejector spring action and see that the firing pin is loose in the bolt.
I would not buy it sight unseen at the asking price. Paying over $350 for an unknown carbine is risky. The barrel should be inspected and gauged. It should be checked for headspace. The gas piston should move freely and look clean. The slide should not jump out of it's groove in the receiver when cycling. The action should pass a static function check. The trigger housing should be snug in the receiver. When the action is 'hung' on the recoil plate, the barrel should be well above the barrel channel in the stock, and should settle to within about 1/16" of the channel when the stock is held with one hand and the buttplate smacked several times with the other. In other words, the barrel should not be forced down to assemble the gun and neither should the barrel lie on the wood.
There's really a lot more, but you get the picture that there is a lot to look for. It would take several pages of this thread to explain them all.

school is in session, all VERY good points INLAND

[wensteph]

I don't think extraction issues will be fixed with grease. There are some good threads on the various carbine forums about extraction and ejection problems. On my first carbine, I ran down the list on one of the threads. First isolated a magazine I knew worked. Wrong or extra power recoil spring can cause problems. Correct spring should measure 10.25 inches. Ordered a new and correct one and problems got better. Seriously cleaned bolt, bolt parts and chamber. Problems got better. Rebuilt bolt with new ejector, extractor and springs and problem went away. Sometimes you need to run down the check list; several things can contribute, none are particularly expensive to fix.