Where can I get hold of an old arcade cabinet

You can try eBay OR

http://www.arcadewarehouse.com/

Your best bet is to buy a JAMMA cabinet - it's an industry standard, and then all you need are the different boards to put into it. You can also stick an xbox or PC inside it, and play mame through it if you're prepared to do some (fairly noddy) soldering.

AFAIK Salamander and Gradius are JAMMA games so, as Rocko says, look for a JAMMA-compatible cabinet with an 8-way joystick and at least 2 buttons and you'll be laughing. Apparently R-Type isn't a JAMMA game though, so you'd need to obtain (or make) an adapter to convert it to work with a JAMMA interface.

My golden piece of advice to you is this - if you're buying a cabinet, make sure it has a working monitor. The monitor is by far the most diffcult part of a cabinet to repair, so if the cabinet already has a duff monitor then you're probably best off avoiding it altogether.

As you've correctly surmised, the MAME-compatible cabinets are the ones that have had a PC fitted (or been modded to easily accommodate a PC) which is running a version of MAME. While technically it's possible to make a cabinet capable of running both a MAME PC and a JAMMA board, usually you'll find that it does only one or the other.

So think about whether you'd be happy with a MAME cabinet or if you'd prefer an original JAMMA cabinet & board before you start.
You *can* get excellent results with MAME which are almost indistinguishable from the original (I use AdvanceMAME), but I've never seen anyone else's machines so I'm not sure if you're likely to find the same with a cabinet off eBay.