Here are some possible explanations. I've listed them starting from the simplest to check.
Hard Drive writable
Make sure you aren't accidentally trying to install either program to an external drive, or any other file mount. I'll admit that I have mistakenly done this once.
Double check that your download arrived intact.
On the atlassian support page, an employee offers this suggestion to someone with the same error as you
I downloaded SourceTree 18.104.22.168 and successfully installed it. Compare your hash to mine.
$ shasum SourceTree_22.214.171.124.dmg
Mac 10.6 Bug
There is a bug in Mac OS X called the dot_clean bug. Take a look at this website for how to fix it. To avoid it, don't save the file to any DOS formatted drives.
Try creating a new user account, and see if they can install it.
3rd Party Filesystem drivers
Make sure you aren't using any file system drivers like fuse.
Bad Hard Drive
Take a look at your root directory, if you see a folder named 'DamagedFiles' it could be a sign that your hard drive is failing.
Scan for bad sectors with a program like Drive Genius or TechTool Pro
Run a hardware diagnostic
Take your laptop to a service provider. They will be able to perform a Netboot which will be able to further diagnose any other hardware issues.
If all else fails, backup to time machine and perform a fresh install. When you restore from Time Machine, all your information will be restored exactly how it was before.
If you happen to have a blank external hard drive, install a fresh copy of Mac OSX to that drive, then hold the c during boot to choose which hard drive to boot from. See if the programs have the same install issues on the fresh OS.
I've been repairing PC's and Mac's for 15 years. Because you are seeing significant slowdowns as-well, my gut tells me that it is likely a combination of hard drive failure, and file corruption.