The latest version is 5.7.1, but having said that, just download and install the previous version.
I would also first suggest to go with the latest update, as this is supposedly the most bug-free version.
If you insist on LR5.1, perhaps you have your original download file from Adobe still in your download folder? Then uninstall LR5.2 and reinstall this "Lightroom_5_LS11_mac_5_1.dmg".
For saftey reasons I would always keep these download files in the latest version, so you could easily go back.
Shortcuts for copy-paste-settings? Are these personal ones of you (e.g. because you use LR in English on a keyboard of another language) or the Adobe default ones?
Your personal ones you need to copy back after every update: from Applications, show the content of your LR App, go to folder "Resources" and put your "TranslatedStrings.txt" into Folder en.lpoj.
There never was a Lightroom 5.1, and unless you have saved the Lightroom 5.0 installer on your hard drive, you can't go back to 5.0 because you can't download it any more from the Adobe web site apparently.
However, the original problem seems like it would not be solved by switching versions of Lightroom.
The original problem is described as
Since I updated Lightroom to the newest version 5.2 my healing brush is not working any more. Plus I can't use the short cuts on my mac anymore to copy and paste settings.
So caten42842801, do tell us more. What happens when you try to use the healing brush. Give us plenty of details. PLENTY OF DETAILS. Same for the shortcuts problem.
I'll bet this has nothing to do with Lightroom versions.
I have a Mac OS X Version 10.9.5 if that helps at all.
After I updated Lightroom and tried to use the healing brush on one of my pictures the brush would only work over the arm of the person but when I tried to use it anywhere else like the face it wouldn't even show the brush.
In turns of the short cuts. I never set any on my own. But I used to be able to just press command-c and copy some or all settings. Now I have to right click on the picture -> develop settings -> copy settings. And the same with pasting them. I only had to press command-v and it would paste all the settings. Now I have to right click on each and every picture and right click -> develop settings -> paste settings. It's so much more time consuming.