Your comments

Those values are way over the top. Here, check the FileRun PHP configuration docs, section PHP file upload handling:

PHP file upload handling

file_uploads            = On
upload_tmp_dir          = /path/to/your/temp/folder/
upload_max_filesize     = 20M
post_max_size           = 20M

Note: Replace /path/to/your/temp/folder/ with an actual path.

“post_max_size” needs to have at least the same value as “upload_max_filesize”. If you go with larger values than 20M keep in mind that your server might have other limitations in place, which might prevent FileRun from uploading file in larger chunks. If having troubles uploading larger files, always set the two settings lower not higher, as low as 2M. If you are sure that your server can receive larger pieces of files, than increasing these two values will allow for a faster throughput.

Assuming you are using Apache as a webserver, did you check the access and error logs? Especially check the HTTP return codes of the server responses in the access log. It seems like a possible caching issue to me. 

As the foldernames have "nextcloud" in them, I checked the Nextcloud app's settings on my Andoid phone. I tapped Settings (hamburger menu) > Uploads and there I noticed that 4 uploads failed and/or waited for restart. As these uploads had trashbin icons next to them, I deleted all 4 from the failed upload queue. That did not clear the "_temp-filerun." folders in FileRun. So I decided to delete the folders manually. Once deleted, they didnot return.
The retries for the 4 failing uploads in the Nextcloud Android app must have caused the "_temp-filerun." folders to spontaneously re-appear after manual removal in FileRun. 

I entered the Control Panel again, selected Configuration > More > Software licensing, and got the e-mail prompt again. I re-entered the same e-mail address and clicked OK. Now I got the FileRun installation is already registered prompt with the client account link in it. Then I clicked the client account link which brought up the Paste received license key prompt. I clicked Cancel to retry the procedure and the 2nd time when I clicked the client account link, I was taken to the FileRun.com login page. On my 3th try, I clicked the OK button in the FileRun installation is already registered prompt and finally got the Paste received license key prompt again. From there on everything went smooth. 
My suggestion is to add descriptive texts in the prompts explaining what to click for what action. As it is now, it is just not obvious to me as new user what to do or what to expect. As this applies to registering a license, you tend to be super careful.

"A while" is in this case 10 mins. before the folders re-appear after being deleted.