I also have a 13" MacBook Air, but it is a Mid 2017 model with a 252 gig SSD (I am using the latest version of High Sierra, OS 10.13.4 (also have a late 2012 Mac Mini with a 256 gig SSD and 8 gig of memory, and also running OS 10.13.4 on it)). How much disk storage did your MacBook Air come with, 128 gig (actually more like 122-123 gig), or 256 gig (actually 252 gig)?
When you say "I only have a sad 8GB on this laptop", that is actually the amount of memory it has. I previously owned a mid 2013 13" MacBook Air with a 252 gig SSD, but it only had 4 gig of memory. The machine ran fine, but the 4 gig of memory, at times, became a bottleneck. So, I sold it in December and got the mid 2017 model, and it has 8 gig of Ram. Definitely a nice upgrade!
Regarding some excellent freeware that can be of assistance with disk cleanup/maintenance/repairs, from a software perspective, here are 3 stellar programs:
, available from here: https://www.titanium-software.fr/en/onyx.html
A stellar performer, it has been around for quite some time, has stellar reviews, and for myself, is essential. (I also have the excellent commercial program Tech Tool Pro that I use "in conjunction" with Onyx). Note that Onyx has a version for each Mac OS, and it is essential to use the correct version. For you, that would be V3.1.9.
, available from here: https://freemacsoft.net/appcleaner/
Another gem! It does an excellent job of finding most files associated with an application that one is deleting/removing. It would have been good to use it when you tried to get rid of MacKeeper (more on that below).
, available from here: http://grandperspectiv.sourceforge.net/
Slick graphical way of seeing what files are on your internal drive, and thus could be used to locate large ones that are no longer necessary.
Next, regarding backups, there are two ways of doing it: use Time Machine (it comes as part of the OS, and is free), or use a backup/cloning program like SuperDuper! (I use it) or Carbon Copy Cloner. Each method has its advantages and disadvantages, but I find SuperDuper! the way to go for 1) installing a new Mac OS, and 2) recovery from a disaster. This thread I started back in November, along with many useful comments from others, would be good for you to read:
Now, regarding MacKeeper, you can use a search tool to try and locate the remaining files associated with MacKeeper. Just put "MacKeeper" in as the search term. The Mac OS comes delivered with Spotlight, but I prefer the excellent freeware program EasyFind, available from here:
Finally, it is possible that the disk directory is somewhat "hosed", and thus is causing you issues with your desktop. You can run Disk Utility from the (hidden) Recovery Partition, and let it Repair your internal drive. However, without a backup, that is risky. Be that as it may, this link describes how to boot to the Recovery HD partition, and what one can use it for: