Decided to go with Discourse as our forum of choice, and opened an account with Digital Ocean. I have successfully installed Docker and Discourse on our server there, however the domain “discourse.dynomotion.com” shows up as an error 403-forbidden up in my browser, so I can’t register and finish my setup. Any idea why that is?
That looks like an NGINX or Apache error page - are you running something else intentionally on the server?
Well, I set up the subdomain “discourse” on my existing host (A2 hosting). I was planning on leaving my existing site there, and use Digital Ocean just as a host for my forum.
Now when I type the IP address of my Digital Ocean server (that runs Discourse), I do see “Installation successful. Register…” - after registering, I was waiting for an email that never arrived, even though I am pretty sure I have entered all the correct SMTP information.
So the subdomain is an A record pointing to the IP address of your DO server, correct?
Having read your reply I realized I had forgotten that - not very unimportant - step 
It is pointing to the DO server now and working. So at least I am only left with the issue of the missing email… I specified the correct server name and port of my hosting company