Don’t thank me.
The man with that huge gun standing behind me is forcing me ;D

Most video’s show avast for Windows.
avast for mac is with some things (slightly?) different.

I’m not a mac tech, but it can very well be that smart scan is only in the Windows version.

I can’t tell by the info you have provided till now if you have a valid copy or not.
The location from where you got it can tell a lot.
Just some legitimate download locations:
https://www.avast.com/free-mac-security
http://download.cnet.com/Avast-Free-Mac-Security-2015/3000-2239_4-75451160.html
http://avast-2015.en.softonic.com/mac

And for the files that couldn’t be scanned…
I would not worry about it if you don’t experience problems with the system.
The amount of files on a system vary all the time.
As you said, cleaning up things, putting things in the trash…
Temporary files that are there when needed and (automatically) removed when they are not needed anymore and such things.

Also, once Avast finds infected files or malware, etc., how should I handle it? Is deleting the best option, or quarantine?
I recommend quarantine. The reason for this is that it is always can be a false positive. If it is you can place the file back from the chest. If after e.g. 2 weeks everything is still working as it should, you can delete the file(s) from the chest. If the file(s) really are harmful, they can't/won't do any damage at all if they are in the chest.