Had a cheap Logitech gamepad before.

And after seeing how half the games didn't work, I bought an X-Box controller. Haven't had a problem since.

I presume it is an old kind of DirectInput controller, as apparently the lower shoulder buttons are merely (digital) buttons, and not XBox360-like analog triggers, right?

Your cheapo Logitech gamepad was probably an old DirectInput gamepad, and yes those can be problematic with newer XInput games.

New Logitech gamepads, the F-series, work fine with XInput games, all the same ones where XBox360 gamepad works. On top of that, F-series work fine also with older DirectInput games, even without XPadder or anything like that.

Yes. I like the idea that the Logitech F710 (310, 510 too?) has that button to swap between XInput and DirectInput, very handy.

I second this, you can usually find it a lot cheaper than a 360 controller. The only problem I've noticed it that sometimes in games the default bindings just suck plain ass, but that's where key re-binding comes in.

Also, the square sticks never really bothered me much. I'm plenty comfortable with it.

Get a PS2 Dualshock and an USB adapter off Amazon.com. The upside of this over the Xbox is the Xbox requires extra software (XBCD) to remap the left sticks due to the analog being the default. The reason is if you're going to use this on emulators so won't accept input from the hatswitch, which is what the Xbox Dpad is normally mapped to. And Dualshock's are cheap, durable, and easily replaceable.

So, I've been thinking of getting a gamepad. One of those Logitech ones that's being talked about so much in this thread as it is definitely advantageous to have one that can work with older and newer inputs without much problem.

That said, I'm not sure whether to go with the F310 or the F510. Is the rumble worth the extra $20? Are there even enough games that support that feature on the PC? I know how rumble feels like on a PS3 and Xbox360 feels like, but does anyone know how Logitech's pad stacks up in comparison?

Alternatively, would the F710 be a good choice? To those of you who may own one: how good is the wireless? Are there any connectivity/accuracy problems I should be concerned about?

If you can get a hold of the "premium" xbox 360 controllers with the transforming d-pad, I would go that route. I have one, and it's pretty much the perfect controller. Of course, you need the wireless receiver, but make sure it's an official Microsoft one. Also, I know that two out of the three games you mentioned in the OP (Psychonauts and Alan Wake) support the pad in question. I haven't played Jade Empire, but I'm sure you can find info out there.

I had a PS3 for a year and the DualShock 3 was my first pad ever. Thought it was fairly fine for such a thing, but didn't really have anything to compare it to. When I changed consoles to an X-Box 360, I initially thought the pad worse, but it grew on me in just a couple of days. I now think its a much better design, and have been using it for all my pad needs since.

Yes, the d-pad is poor, but it's seldom a problem. Fits my hands snugly and the stick placement just works better for me. My hands are somewhat on the larger side, which may be a factor.

No.
If you have ever used a powersander in your life you know how miserable this feature is.

Speak for yourself. Rumble feedback is great, aside from the immersion it helps when you take damage or the like.

It's pretty comparable, but I don't know if it's worth 20$. It is used mostly in games ported from console or released on multiple platforms.