Upon entering the game the circular main menu just spins wildly out of control until I press some random key or pad-button, and then when I try to configure keyboard or pad by pressing enter in the controls menu, the game just locks up. Any suggestions, anyone?

Same thing is happening to me.

Yeah, it's happening for me, too.

I must be lucky because i'm not having that or any other serious problems with TR1, my problems are with the cutscenes not working with TR 2 & 3.
Here are some of my PC specs in case it might help - Intel Core 2 Duo, ATI Radeon HD 3450 and Win XP Pro SP3.

So far it seems to run fine if I unplug my 360 controller, but using the keyboard for this game is awful.

I'm speculating that the problem is a hypersensitivity to the joystick. I think that a possible solution might be to map joystick controls to keys. I don't know of any good programs for that, though, and Google isn't giving great results. The best thing I've seen is JoyToKey, which doesn't look very promising and has a terrible interface.

Or, I guess, alternatively, we could get a simpler gamepad that doesn't have a joystick, or maybe a better joystick (I'm beginning to think that the 360 controller isn't exactly great quality, and I'd be interested in an opinion of someone that knows better than I do).

Got pretty much the same exact specs as you, so that can't be the issue here.

I used to Xpadder to manually map controller controls to keyboard, and I remember it working pretty well.
But I'm wondering if that would work, considering it's autodetecting the gamepad? Mapping to other buttons won't change that I don't think.

You can fix the menu spinning thing by editing the file "dosboxTR.conf" and setting "joysticktype=none". However, you won't be able to use a gamepad natively in DOSBox if you do this.

You can then use something like xpadder if you still want to use a gamepad. Since the game has no analog controls at all that works just fine.

Setting timed=false from the same file, dosboxTR.conf, also helps quite a bit. It makes it at least usable, but it turns out that TombRaider, or maybe DosBox in general, only accepts input from A, B, X, Y, and the triggers, and the triggers are both the same button, JOY6. The left joystick does work, but is not configurable and is still overly sensitive.

I'm reluctant to spend the money on XPadder, personally.

Bone "triggers" on 360 or clone controller are the same input axis. That's why I keep PS2-styled controllers for older stuff, they have real buttons.

Doesn't DB allow configuration of the joystick in the settings?

Possibly, but I'm pretty new to DB so I don't really know how yet. And Gog games use self-contained versions of DB, which keeps throwing me off. I'll keep experimenting.

Changing joysticktype to fcs in dosboxTR.conf also allows the gamepad to be JOY5.

So far I have the game at the very least playable on a 360 controller, with "Step Left" and "Step Right" not usable (except on keyboard) and I still have to press Escape to bring up the menu.

Update: There is the "mapper" that can be brought up by Ctrl+F1, but using it seems kind of buggy. After I bring it up, I have to exit the program before I can do anything else. But more importantly than that, it only shows buttons 1,2,3,4 (A,B,X,Y), it doesn't show 5,6 (back,start), or 7,8 (LB, RB). It kind of leads me to wonder if DosBox just doesn't recognize those.

I think changing the lines in dosboxTR.conf from

joysticktype=auto
timed=true

to

joysticktype=fcs
timed=false

is definitely the way I'm going to go, myself, though it's not a perfect solution. The joystick is still a little annoying; sometimes it keeps moving when I let go unless I nudge it down just a little bit. And I have to make sure that the joystick is absolutely straight up or else it'll move to the left or the right. I had a similar problem with Resident Evil 2 on Dreamcast.

I also still might get a gamepad that doesn't have a joystick; that would be good for other games, too. But I tend to think that it wouldn't be able to do more than four buttons with DosBox, either. I am actually wondering if I would just like to play it on the keyboard, though that would take a lot of getting used to.

Also, as a side note, I discovered you can have it start out windowed instead of fullscreen, which I personally think is awesome (combined with changing the glide.ini resolution setting to 1024).

A little piece of software called xpadder solved my problems.

http://en.wikipedia.org/wiki/Xpadder

Xpadder is a good work around, but a true solution would be much better.

Also, getting an old Gamepad might be helpful for old games. This one has been very helpful for me:
http://www.amazon.com/LOGITECH-Gamepad-Comfortable-Programmable-Buttons/dp/B004NQTR5Q/ref=sr_1_7?ie=UTF8&qid=1338588566&sr=8-7
The Xbox 360 controller always has weird problems with the analog triggers. That Logitech one doesn't.

That Logitech looks good, I've been thinking of getting either that or something similar to it.

JoyToKey works well enough, I've discovered. I wasn't confident when I first looked at it, but after trying it out it works just fine. I was going to put a tutorial on here for how to use JoyToKey (starting a new thread), but Gog won't let me because the post is too long.

adanedhel728 and Lafazar really helped me with this. Just disable the joypad by editing the "dosboxTR.conf" and setting "joysticktype=none".

Then just download Joytokey and map each of the default tombraider keys to each of your gamepads desired buttons.

http://electracode.com/4/joy2key/JoyToKey%20English%20Version.htm

Been playing with my gamepad now for hours and haven't had any problems.