OK, I've been down this road & here's what I found out. fyi-my car is an original bayonet pistol grip car.
1. The factory linkage (where shifter bolts into linkage) is sloppy.
2. I tried to drill some holes sideways & put two small bolts/nuts through the linkage, but it didn't help. Looking back, I think I drilled the holes through the "split" in the bayonet, so I didn't really bolt anything together.
3. I've seen tack welds on other cars where the shifter meets the linkage to take out the slop. That would work for sure, but the shifter/linkage would have to go out the bottom of the car to remove the transmission/clutch...not the worst thing, just a fyi.
4. No, you cannot just install a bolt-on (or bolt-through) shifter unless you use a bolt-through linkage (I think that's the way the aftermarket ones are???)
5. Yes, I've taken those bayonet clips out, but not replaced them. With the shifter out, they pretty much just lay inside the linkage. I don't think replacing the clips would take out any slop though.