It looks like the original apk was damaged.
I tried to look at it using 7zip on Windows - and while showing the content, 7zip reported problems with a bunch of files (it was unable to extract them).
Kostya, - I think it is a good decision on your part to switch to https. This doesn't protect against the files being replaced on the website, but it somewhat protects against a man-in-the-middle attack.
As for my semi-joking mentioning of malware, - the way Android was handling this was weird.
As I wrote earlier, the package installer didn't give any error, any warning, nothing.. It just died (exited) silently. (BTW, there was no usual confirmation screen that this application will replace the existing one...)
But on top of this, - both ES File Browser and Android's built-in "Files" were dying when I was opening the "Downloads" folder where the damaged apk was residing. (Such a behavior could've been consistent with the malware preventing tools from accessing the folder it was residing in.)
The only way I was able to do anything with this folder, was to delete the file by using Opera's interface for the downloaded files.
It is very-very strange that ES File Browser and "Files" were freaking out while looking at the folder (not even at the file itself). Does Android has some sort of automatic analysis of the files residing in the folder?