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For your lag issues, the game uses DOSBox with CPU cycle setting of 8,000, which, isn't enough for high res. When you load up the game, hit Ctrl F12 a couple of times, or as many times as you like until you get the speed you like(so it becomes less laggy). I normally go with CPU cycle setting of 11,000 or so, runs a lot better.

As for your issue with the traps and doors, I've never seen that before. Only thing I can think of is it missing some type of file or game file. Did you do anything with the creatures.txt file?

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It's an issue with the game engine. It can only handle so many moving sprites, whatever kind they may be. You might have noticed that some of the animations also disappear - graveyards don't produce mist, workshop tiles lack workstations and look like plain floors, training room tiles lack training faciliites, torture rooms lack torture tables.

It's a bug to do with the fact that sometimes enemy AIs can end up putting multiple traps - sometimes tens or hundreds of them on the same tile, often quite illegally (e.g. on a room tile - or perhaps the auto-build of room tiles was allowed to put it over a trap?). This ends up massively over-consuming the number of "present" animations in the level, leading to hitting the "max" when one shouldn't be. Often, triggering off these enemy trap tiles can end up allowing you to play normally once more... or crashing the game. You never quite know which.

It may be a case that you simply have to complete the level faster, if you can.

I noticed that I never had this problem where there was not an enemy Keeper, only heroes. Nor did I have it if the enemy Keeper was unable to lay traps.

I also noticed that this bug did not seem to happen in the latest version of Dungeon Keeper, only Deeper Dungeons (which, if I remember rightly, had sort of forked from an earlier unpatched version of DK, and had the bug still uncorrected.) Mind you, in standard Dungeon Keeper, the only level which has even the slightest chance of lasting long enough to hit this bug is the very last level, Skybird Trill, and you don't necessarily even have to fight the enemy Keeper at all if you can defeat the Avatar and his cronies twice. Other standard DK levels are too short to actually ever hit this bug.