Hello,
I purchased EECH online but I can't seem to download it either using the GOG downloader (freezes) or using my own '"Free Download Manager" Windows utility or even using Mozilla Firefox.
I tired several times.
How many times can I download the game and what should I do?

You can download it as many times as you want - no download limits. If I have to guess, I'd say it's probably some routing issue with your ISP. Make sure you contact GOG support - they will help you find the problem, whatever it is, or at least figure out a way for you to download the game while the problem isn't fixed. But again, I doubt it's a problem in GOG's end - most likely some routing problem between your ISP and GOG.

Did you get the game downloaded? If not try checking your firewall setting and turn off your antivirus, also check your screen saver and power setting to insure that the download isn’t being interrupted, and also close any opened windows and programs. Contact your service provider and have them run a check on the line, I found out that I wasn't getting the service I was paying for when I had mine checked. It could also be a modem issue.