COD 4 crashes on the desktop during mission ‘Shock and Awe’

I've never tried online play myself.

The brilliant guide RTSC Startopia has a few notes on its starting page. In particular, they say:

• That online play, different from LAN play, doesn't work well. (Yeah, you figured that out yourself.) Meaning it's probably caused by desyncs due to the higher latency, something many older games struggle with.
• That the autosave helped them after the frequent crashes.
• That there's a rumour that the autosave itself causes the crashes and that disabling the autosaving by write-protecting the file might help. (Could this explain why it's almost after ten minutes for you?)
• Do not try to save. Do not pause. Do not open the menu. (More indicating that desyncing is the issue at hand.)
• That you shouldn't get frustrated by crashing multiplayer games, because it will just happen a lot. (Probably the most helpful tip, but also much easier said than done. :))

I did some quick googling, and the ioquake3 project seems to have a working client you can use with ID's own data files.