Restoring apps and preferences to a clean install
Is there a way to do a clean install but keep all of my apps, preferences, etc.? Could I just make a Time Machine backup and restore from it?


Short Answer : Yes you can. OS X has an App called "Migration Assistant" that is located in "/Applications/Utilities/Migration Assistant.app", which can easily restore user-chosen stuff from a backup or an external HDD or external Server Partition or a Thumb drive if you wish to do so.

The Long Answer :
As a clean install is always helpful, don't try to simply run away from your problems. OS X can be a very sensitive System and Mavericks has its problems.
(And thank god there is Stackexchange for every Problem that might occur)

I would consider using certain tools to clean your mac and make sure it is checked carefully. Some common Apps are OnyX or CleanMyMac.

Also try creating a whole new user and compare the performance with your own account. Using this way you can find out if there is something in your account that makes your machine so slow or if it is the overall System.

Also try using the PRAM and SMC -reset, Boot your mac from a recovery partition or CD/DVD to repar the Disk if neccessary or to check if everything is ok.

If nothing changes during all your steps, a clean install is your last resort. If even this does not seem to fix the problem it won't be a software issue and you should take the Computer to an Apple Store for Hardware checks. (you can use the build-in hardware check first if you want to save some time)