Grind has an influence within the context of each coffee maker, but not much across them. On average, despite using coarse grounds, a French Press will produce more-caffeinated coffee than a drip pot will with medium grounds. Turkish coffee will have the most, but it doesn't have as much to do with the grounds as the fact that you'll actually be ingesting a fair amount of the beans themselves.
But yes, a French press with finer grounds will produce a coffee with higher caffeine content. Doesn't mean much though, because you really shouldn't be going below a medium grind with them (and even that's a stretch), because the mesh plunger won't filter that out. You'll get grainy coffee.
As for filters, it's really a toss-up. It's much better for your cholesterol to use paper, the coffee tastes better if you use metal. Caffeine won't be too meaningfully different in either (I'd guess it's less than a 5mg difference). It's not bad, per se, to be using metal.
If you're brewing decaff (for some crazy reason) though, use paper. The caffeine in coffee helps offset the cholesterol through a process I don't understand. You don't get that in decaff, making it much worse for you.
Of course, coffee is more than just caffeine and cholesterol. It's actually a surprisingly healthy beverage for the body, and there's a significant decrease in risk for a few different diseases from its consumption. And the most common con for it in public mind, blood pressure, actually doesn't hold true--long term studies on hypertension have not revealed any link between coffee and the disease.
Good news for me.