How do I end a level (Or trigger a door) when a boss enemy (e.g. Mastermind) is killed?

There are 2 main methods:

1) you place your map in E3M8 map slot, since here the death of the spider mastermind is hardcoded to end the level. This method, of course, has several limitations. You have to use Doom1 wad and you are forced to place your map in that slot.

2) This method is far more flexible than the previous one, but your map has to be in Doom-in-hexen or UDMF map format. If this is not your case, you can download a wad converter from the realm667 site (it is a very light and easy to use software). 

Once your map is in one of those format, you will have acces to a new section in the thing tab of your map editor (but I'm not sure if SLADE supports the previous formats, so maybe you have to download also Doom builder X or GZDoombuilder from their sites). The new section will allow you to give to a thing an action (exactly like the one of a linedef). This action will trigger once the monster is killed, so assigning an "exit level" action to the spider mastermind will end your level, or a "door open" action will open the tagged door.

I strongly suggest to use the second method.

To add to Simomarchi's post, in the original Doom engine there is very limited functionality to have something happen when a monster dies.  There are a very small number of hardcoded instances (i.e. it only does certain things with certain enemies on certain level numbers) that you have to very specifically design for, but for the most part basically you can't really make much happen when a monster dies.

In much more modern source ports though, e.g. GZDoom, especially if you're using the UDMF map format, all those limitations were removed and there are far more options.  You can have basically anything happen when a monster dies.  The method Simonarchi outlines is how you'd do it.

Depending on how the fight is "staged" it might even be possible to make this work in boom format (entirely without dehacked), regardless of mapslot... But since you didn't specify which format you're using it's nigh impossible to even give a decent answer.