Should I run Chkdsk at all?

If you have an external drive, create an up-to-date backup of all your important data (or at least as much as possible given the drive problems) and then run chkdsk with the /r option (same as checking the "scan for and attempt recovery of bad sectors" option in the GUI). Note that this might take hours, depending on how bad the situation is. I think marking the bad sectors ought to at least enable you to keep this laptop running for a while longer without complaining as much.

Yes you can run chkdsk /f to fix the errors on entire drive and there are very small chances to lost your data, for that case you can backup your data. For backup you can use external HDD or DVD's.

Also you have to run defragment utility provided by the Windows which will help you increase the speed of your system. Download the CC cleaner and scan your system for the junk files and delete them it will also help you out to make it fast.