Ok, it's been ages since I've touched anything Goldsource related, so here goes.
First up: you need to go through your config and change some things. First up, the mod folder. This needs to the TS folder. Same as "Place compiled maps in this directory before running the game", which you'll need to change to the maps folder of the TS directory.
Strictly speaking, if you compile a map, it should now dump the compiled BSP in the TS map folder. You can then manually launch TS and load the map via the console.
You can get Hammer to load the game though: what compile options are you using, normal or advanced?