I recently got a Mec-Gar magazine for my argentine 9mm, but it still has problems.  It has three problems.  First, the slide almost never locks back after I fire the last round.  Second, it often only partially ejects the casing, and the casing gets wedged between the ejection port and the chamber of the barrel.  Sometimes, IIRC, it feeds the original spent casing back into the chamber.  Sometimes it ejects, or partially ejects, the spent casing but fails to feed a new round.

It seems to me that all these problems could be the cause of a recoil spring that's too strong, but since I don't know much about handguns, I'm not sure.  What do you all think?

The recoil spring might be the problem, but first examine the ejector and make sure it's not broken off a little and also check the extractor to make sure it has a good clean, "bite" and pivots or flexs (depending on whether it's external or internal) when it is engaged.

The recoil spring is intended to do just that absorb recoil. If a limp gripping techniques is used your arms absorb the recoil instead preventing the gun from cycling.  First try gipping it firmly. If that fails then look to weak ammo or lighter recoil spring.

I just tried shooting a magazine while holding it very solidly, and I had no problems.  I think that before I was letting my hand absorb too much of the recoil.  Of course it was only one magazine.  I'll have to wait before I try it again, but I definitely will as soon as possible.  I'll try holding it limp-wristed and then solidly and see what happens.

But is that normal, where the gun doesn't cycle properly if it is not held properly?

Some pistols are more forgiving than others, but it is normal for limp-wristing to cause problems.  Your grip provides the "anchor" for the frame to keep it in place as the slide travels rearward to extract, eject and chamber the next round.  If the frame moves too much to the rear, it is moving in the same direction as the slide as it begins the cycle.