Odds are it isn't a Remington at all, but rather an Ithaca Model 37. To my knowledge, it was the only bottom-eject pump shotgun ever made. As for value; any gun (or anything else for that matter) is worth whatever another is willing to pay for it.
If it's a 16 gauge shotgun, the shell length is 2 3/4 inches.
Also, there are no serial numbers on this shotgun but there are a series of dates, 1900, 19001 and it says Browning patents - manufactured at Remington arms - How can I date this gun?
Ammo, parts, dirty
Go to the bottom right and left click on the eject symble and click on the option
part failure, dirty, etc.. Take it to a gunsmith
Yes, if you eject it into a roaring fire.
My first guess would be that it needs a real good cleaning, this is a gas operated and not a recoil action. They haven't made them in about 50 years there might be a bit of junk in there ;) Remember this is also Remington's first gas operated shotgun it and there have been a lot of improvements it won't be as forgiving as modern versions.
The gun does not always cycle. It also will not always eject the shell. These can both be potentially dangerous situations.
Try cleaning it, different ammo. If that doesn't work, take it to a gunsmith.
You need the services of a gunsmith. There are about a half dozen or so reasons if not more.
You need to cycle the action for every ejection.
Any of the following could be the culprit; dirty, ammunition, springs. Take it to a gun smith.