Trem uses the Q3 engine. Q3 was meant to run on 1999 hardware. Go figure.
Well.... Having said that it's not quite *that* simple as many trem maps are far more complex than those of Q3, etc. But really it should run at a decent framerate any recent (i.e. 3 years old or newer) machine. I don't know what's causing the low FPS on HL1 - I got 20 FPS at 800x600 on a 400MHz box with 64MB of RAM and an 8MB graphics card. Half-life 1 was released in 1998 and written to run on the hardware of that era. Sounds like you don't have the 3D drivers for your graphics card installed.
I get 90FPS at 1280*1024, maximum quality settings, with an athlon 64 3200 (2GHz), 1GB PC2100 and a geforce5900XT AGP graphics card. Your hardware is not the fault - it should outperform mine by a long shot.