Yes. As long as you're chaining these "combo" moves together without interrupting using a non-combo move, the hits will continue to count up even if you switch which specific skill you're using.

A really good choice appears to be to use Way of the Hundred Fists as your second strike, since the second strike of that one hits multiple times but the other ones do not.

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.