Just about the only thing you can do to speed up the skill switching process is to right click on the skill you'd like to change. By doing this you'll jump directly to the skill selection for that key, bypassing one page of the menus you'd normally click through.

This is the only way to speed up the process of changing skills. There is no other way to automate or hotkey skill selection in Diablo 3. It must be done through this menu.

Any 'out of the box solution' would require, at a minimum, the use of some form of third party macro or automation software, a clear violation of Diablo III's Terms of Use. While such violations are not always detectable or acted on, the possibility of action against your account and revocation of your license to play would give me extreme pause in considering any other solution.

This is easily soloable although which class you play and your playstyle can affect your success. As you've probably figured out, killing things and fighting in general is the wrong approach. To complete the Crumbling Vault and any of the timed achievements, there are only a few things which can help you:

Luck:

• As the dungeons are generated pesudo-randomly, if you get a bad layout, your chances of success could be greatly lessened. There's a bit of a pattern to the map layouts that you can sort of pick up (possibly even picked from a set of pre-defined maps? I only mention this because I've seen some identical layouts when re-rerunning certain dungeons).
• Running into some stall champions is also a danger to your success as wallers and jailers can stop you in your tracks and their presence is entirely random.

Speed:

• Movement speed perks are helpful.
• The use of your movement ability is also invaluable. To support the use of your movement ability, having resource regen and resource improving equipment and abilities is helpful. To this end, also go in with full resource.
• Don't stop. Avoid combat and run past everything. Having minions/allies and followers out before you enter the timed section is helpful because they can draw agro away from you.
• If you do get stuck in combat, for your primary and if needed secondary attack ability, something with knockback is invaluable as it can clear a path for you very quickly. It is also preferable that this attack have some kind of AOE to clear larger space at one time.

Power:

• High health, dodge and armor can make it easy to tank your way through without worrying about your health. Buffing these can greatly help survivability. I find dodge more useful as attacks that miss don't affect durability or require healing.
• High attack power can really help speed you up as being able to one-hit the enemies will make certain that moments stuck fighting will be brief. Note that your goal isn't to kill or even fight anything so it might be best to neglect attack if needed for these types of challenges.
• Being higher level helps a lot.
• Playing with a dex or strength focussed class makes a huge difference too. With a level 22 monk, I ran Crumbling Vault on my first try in under 2 minutes. I buffed dodge and ran like crazy, taking only 2 wrong turns along the way. Playing as a witch doctor is not helping your chances so you may need to rethink your equipment and abilities from what you usually use. For int classes, you may need to ignore your Int for these challenges and take the hit to DPS in exchange for Dodge/Health provided by Dex and Strength.
• In order to maximize damage and defense, learning the way that the game calculates them is very useful.