[RPG] How to get mic-shy players invested in roleplaying over Skype

Usually I either describe the dungeon verbally, only drawing out the rooms that the players interact with, or I draw the overview of the map on another sheet of paper. I think the problems you're having are a symptom of using an online table.

Here are some ideas.

1. Do the dungeon verbally .. err... textually. Describe rooms to the players. Load them up as needed.
2. Don't put treasure and monsters on the map in advance. Add them when the players go into the right room. That should help with some of the Fog of War tedium.
3. Treat the party as a single entity. Instead of moving the fighter, then the rogue, then the wizard, then the cleric, just move the party token. When an encounter happens, determine more specific positions.

I'd also like to respond to this part:

my players start leading the monster into other parts of the map, sometimes making the long encounter way longer. The rooms for encounters lose their focus, and the preparation feels wasted since, well, I designed them for the encounter.

That's okay! You're not running a video game. You're running a pen and paper roleplaying game, albeit facilitated by an electronic table simulator. One of the strengths of P&P RPGs is that the players can go anywhere and do anything that the GM can react to instead of being cut off by what the designer/programmer planned for. The fact that the players wander into another room that you thought would be offscreen is fine. It means the players have choice and can act on that choice, even if it makes their battleground a little less interesting.

I play online almost exclusively these days, using Mote. I DM two campaigns, and this particular issue comes up often enough.

First, you have to keep things interesting. Try to design combats with more than just "attack roll -> damage roll". It's not a simple task, but it's really important to make combats interesting. Add some ranged enemies, healing enemies, mages, etc. Throw in environmental hazards, or effects which occur between turns. Give the PCs a time crunch to win. Give them a reason to be concerned.

In terms of process, I have found that the method of forfeiting turns does help. If they take too long, they lose their turn. However, I found that rewarding them for taking swift actions is far more effective. I give my players a +1 circumstance bonus per five levels to any rolls they make during their turn if they begin taking actions within fifteen seconds. I call this bonus "seizing opportunities." It works phenomenally. That, plus some recommendation for the players to plan actions during someone else's turn, means that they are likely to have their plans ready and in motion within those fifteen seconds. The bonus is big enough to be worth striving for, without being so big that it's not worth sacrificing for the occasional tactical overview. Players are much more likely to confer with each other this way. This has sped up our gameplay immensely.

You can adjust the bonus, time limit, and everything else to whatever works for you. I just insist that rewarding players is a more effective motivator than punishing them. It also makes them feel good and have more fun. That is the most important thing.

EDIT: Now that I think of it, this is also a pretty effective way to speed up offline tabletop combat, for anyone else who might be doing one or the other or both.

EDIT EDIT: One other thing I should mention is that when playing through an online tabletop you usually have computing tools available. Take advantage of them! The human brain is great at lots of amazing stuff, like creativity, imagination, reason, and emotion. One thing it has a hard time with is mathematics. Fortunately, computers rock that stuff. It helps to shave more time off of a turn if you have as much action content (rolls, initiative order, etc.) hard-coded beforehand. Dice macros are a simple example.

On a non-computing basis, it also helps to have actions listed out in front of the player. I know there was a lot of hate around 4e's action system, but having a sheet of actions clearly delineated in front of the player was a huge advantage for newbies. Having some clear options in front of them helps remind them of what they can do, and thereby lets them make a decision sooner.

Naturally, you do not have to be as hard-and-fast as 4e. You can even go as simply as reminding the player of what they can do in your game, or give them some soft suggestions.