Here is a comprehensive list of things to check and try for anyone experiencing difficulties installing JuiceboxBuilder-Pro (or JuiceboxBuilder-Lite).
I will update this list if/when any new tips are found.

#1 Check out the Install Issues in the JuiceboxBuilder User Guide in case any of the notes describe the problem you are facing and help.

#2 If using MacOS Catalina (10.15), then please see the additional steps required to install JuiceboxBuilder-Pro here.

#3 Make sure that you are using the latest version of JuiceboxBuilder-Pro.
You can check the latest version on the Version History page and you can check the version that you are using by going to 'Help -> About JuiceboxBuidler-Pro...' from teh drop-down menu at the top of the program.
If necessary, full instructions for downloading the latest version can be found on the Upgrading Juicbebox support page.

#4 Make sure that you have extracted the Juicebox-Pro zip file to your hard drive (rather than just looking inside the file with a zip program) and that you are not running the 'JuiceboxBuilder-Pro.air' file from a restricted file system or network drive.

#5 Make sure that the Juicebox-Pro and 'JuiceboxBuilder-Pro.air' files are not corrupt.
If you have been able to extract the Juicebox-Pro zip file without any errors, then the chances are that your files are OK.
However, you can check the MD5, SHA1 and SHA256 hashes of the current Juicebox files here.

#6 Make sure that you are using the latest version of Adobe AIR.
If necessary, the latest version of Adobe AIR can be downloaded from Harman's website here.

#7 Check out the "Troubleshoot Adobe AIR installation" help pages below in case the problem is with Adobe AIR (rather than with JuiceboxBuilder-Pro).
Mac: https://helpx.adobe.com/air/kb/troubles … ac-os.html
Windows: https://helpx.adobe.com/air/kb/troubles … ndows.html

#8 Try using an older version of Adobe AIR.
JuiceboxBuilder-Pro requires Adobe AIR v2.0 or later so you could try a version such as v20.0.
Unfortunately, it is no longer possible to download older versions of AIR from the Adobe archive download page so finding older versions might be difficult. If you find a link to an older version of AIR from a source other than Adobe, then please download and install at your own risk. (Some users have reported success in using older versions of Adobe AIR which is why I list this as a suggestion.)

#9 Try completely disabling any third-party security software that you might have installed (e.g. Avast or AVG), including real-time protection and ransomware detection, before attempting to install JuiceboxBuilder-Pro in case this may somehow be interfering with the JuiceboxBuilder-Pro installation. (Please do so at your own risk and be sure to disconnect from the internet first.)

#10 Check that your system clock is accurate (and resync with a known time server if necessary) before attempting to install JuiceboxBuilder-Pro. (One user actually claims to have had to wind back his system clock 2 years in order to install JuiceboxBuilder-Pro. This seems to have been an isolated incident and no-one is quite sure why this was necessary but it might be something worth trying if nothing else works.)

#11 Try the complete uninstall/reinstall procedure below.

(1) Uninstall your current version of JuiceboxBuilder-Pro (if you have a version installed).
Mac: Delete the 'JuiceboxBuilder-Pro' file from the Applications folder and empty your Trash.
Windows: Use the uninstaller in the 'Control Panel -> Programs -> Uninstall a program...' list.

(2) After uninstalling JuiceboxBuilder-Pro, please check that there are no files are left behind (even from a failed installation) by manually deleting the following folders (if they exist) from your hard drive:
Mac:
/Applications/JuiceboxBuilder-Pro.app/
/Users/Username/Library/Application Support/Adobe/AIR/ELS/JuiceboxBuilder-Pro
/Users/Username/Library/Preferences/JuiceboxBuilder-Pro
Windows:
C:\Program Files (x86)\JuiceboxBuilder-Pro
C:\Users\Username\AppData\Roaming\Adobe\AIR\ELS\JuiceboxBuilder-Pro
C:\Users\Username\AppData\Roaming\JuiceboxBuilder-Pro
C:\Users\Username\Documents\JuiceboxBuilder-Pro
You may need to show hidden files to find some of the above locations in Windows File Explorer.

(3) Also, please follow the procedure above (Steps #1 - #2) for JuiceboxBuilder-Lite, too (if you have it installed).
JuiceboxBuilder-Lite and JuiceboxBuilder-Pro can both be installed and run side by side on the same computer but, being that you are experiencing problems, it might be wise to completely clear your system of all JuiceboxBuilder files before trying to reinstall JuiceboxBuilder-Pro.

(4) Uninstall Adobe AIR.

(5) Reinstall Adobe AIR.
You can download the latest version of Adobe AIR from here.
Windows: To install Adobe AIR, instead of just double-clicking the 'AdobeAIRInstaller.exe' installation file, right-click it and select 'Run as administrator'.

(6) Reinstall JuiceboxBuilder-Pro (making sure that it is the latest version) following the instructions here.

#12 When using JuiceboxBuilder-Pro, do not load a gallery from (or save a gallery to) an online synced file-sharing folder (such as Google Drive or Dropbox).
JuiceboxBuilder-Pro creates a temporary folder (named '.tempGallery') whilst a gallery is being created or edited. This folder is automatically deleted on closing JuiceboxBuilder-Pro.
However, the use of an online synced file-sharing folder can prevent JuiceboxBuilder-Pro from deleting this temporary folder.
If you run into this problem and find that the 'New Gallery...' and 'Open Gallery...' buttons on the 'Start' tab become unresponsive, then just close JuiceboxBuilder-Pro and delete the '.tempGallery' folder manually.
On a Windows system, the '.tempGallery' folder is located here:

C:\Users\Username\Documents\JuiceboxBuilder-Pro\.tempGallery

Note for Juicebox-Lite users:

The notes above refer to Juicebox-Pro/JuiceboxBuilder-Pro but apply equally to Juicebox-Lite/JuiceboxBuilder-Lite.
If you are having problems with JuiceboxBuilder-Lite, then just follow the suggestions above but replace any instances of 'Juicebox-Pro' and 'JuiceboxBuilder-Pro' in any paths with 'Juicebox-Lite' and 'JuiceboxBuilder-Lite' respectively.