Is there some way I can compile just Pidgin, without all of these protocols?

Yeah. There are actually three ways to do it.

1. Use the --with-static-prpls option to ./configure along with --disable-plugins. This will let you choose which protocols are compiled in with Pidgin, but you will not be able to use any other protocols or plugins. You'll need to compile any protocols that you specified in --with-static-prpls. Oscar is the default.
2. Use the --with-dynamic-prpls option to ./configure This takes a comma separated list of protocol names.
3. Change to the directories for the protocols you want to compile (e.g. libpurple/protocols/jabber), and run make from there. Then after you've compiled all of them, change back to src/ and run make pidgin

Can I run Gaim on IRIX?

Sure you can - the amount of effort involved depends on which compiler you use; there is however no official support.

GCC (tested with 3.4 on IRIX 6.5.29): All but one necessary changes were included in the official sources. You will notice GnuTLS will not be included by default, even if the necessary includes and libraries are available; to add SSL support, open the configure script in your favourite editor and replace all instances of "-lnsl" by "-lnsl -lgnutls". Ask no questions :-) it Just Works. Mozilla-NSS wasn't tested.

MIPSPro CC (tested with 7.4.4m on IRIX 6.5.29): Less trivial for obvious ANSI-tightness reasons. There's an unofficial patch for 2.0.0 beta3 available at ​http://jesuschrist.be/gaim/gaim-2.0.0beta3_irix.patch which applies (amongst others) the SSL hack described above. You will need to configure WITHOUT the Zephyr protocol plugin.

NOTE: Jabber crashed gaim when using TLS. This is a bug inside GnuTLS that might or might not be fixed by the time you read this.

Can I run Gaim on HPUX?