Would reformatting my Mac and restoring the latest Time Machine backup be likely to improve its stability?
There could be various causes for a Mac that runs slow during certain times:


too many applications running on the system (launched at login or explicitly by the user)
inadequate RAM
inadequate free space on the drive
file permission issues
corrupted files or configuration
updated OS and applications that need faster hardware, and so on


For all these cases, reformatting and restoring from Time Machine will not help.

If you'd like to clear caches and logs and repair permissions, there are far quicker and easier ways to do it than a time consuming reformat and restore from Time Machine. Make sure you have a backup and then try some of these utilities (you can also search for "OS X cleanup" to find other utilities):


OnyX
MainMenu
Cocktail
CleanMyMac 2


If these do not help:


if you're running OS X Mavericks, check the Memory Pressure information in Activity Monitor - this will tell you if RAM is inadequate for the applications you're running
try trimming down the number of applications that are running. Start with items launched at login (go to System Preferences > Users &amp; Groups and then to the Login Items tab)
free up disk space (if it's quite low)
using Activity Monitor, check which applications use a lot of CPU and either remove them or reinstall them afresh (trash the application with a proper uninstall utility like AppCleaner so that its preferences and other files are also deleted and then reinstall it)
as a last resort, you may have to do a fresh install of the OS, followed by a fresh install of your applications, followed by a restore of only your data from the Time Machine backup (do not restore the entire applications folder or the home folders since that would bring back corrupt preference files, if any)