well I can speak for Gameshark, it is physically thicker than the Sega brand memory cart. Yes there is a known issue with the cartridge slot, thick carts don't help the problem.
For PAR I honestly cant say for sure since I have never used one personally but assumed it was like the Gameshark.
The Gameshark is literally a rebranded PAR. It lacks the RAM expansion stuff because that didn't come out until later versions of the PAR. The software is slightly different and some have com ports on them, but that's about it.
I think it is because it is just for memory cards and PAR devices, so low market potential vs a cart based system with higher base of people that would buy.
The Power Memory cart launched with the system. I'm pretty sure they thought %10 of users would just buy it, put it in there, and just leave it. The other stuff was obviously accommodated for in the original spec, but I don't think they seriously thought it would happen.
I wish they made cartridge games on the Saturn. Is the system capable of running a game from the cart port?
Yes, it is. The PAR boots from the cart slot, for example. Its never been done though.