This is roughly about 1400 years ago, in present day Mecca/Medina metro areas. Did they have running water? how often did they bath? was there a concept of regular showers? Did they have proper bathrooms, or was it like the rest of the world, in terms of going in a ditch?
As far as I'm aware the hadith are likely to contain the most exhaustive recordings of hygienic habits of Arabia. They were not recorded "in Muhammad's time" and as a result there are all kinds of problems with using the hadith as a source, but it strikes me as being unlikely that where people were pooping would have changed all that much in the 200 years that passed between those times.
In answer to some of your questions (NB: this applies to the Hejaz, northern Arabia, around modern Jordan/Syria would have been Romanized to the point of having bath houses and is not a topic I'm particularly familiar with):
Did they have running water?
In the sense of plumbing? No. Water would have come from wells or cisterns.
how often did they bath? was there a concept of regular showers?
It depends on what you mean by "bathe." The full submersion of the body in water would have been unusual. There just wasn't enough water to do that. Nor is this particularly strange, historically speaking, there's a recent /r/askhistorians podcast episode that similarly describes the lack of full-water bath submersion in the early US.
That being said, highly regular washing is an integral part of Islamic ritual. I'm not familiar with accounts of pre-Islamic washing habits, but nor am I familiar with any suggestion that Islamic strictures to wash were particularly onerous or controversial.
Did they have proper bathrooms, or was it like the rest of the world, in terms of going in a ditch?
The hadith do reference "bathrooms", but I'm not sure what this would have consisted of.
I'd encourage you to read and search through topics that interest on this site, with some sample searches below:
http://sunnah.com/search/?q=defecate
Thank you for the reply! I've found the following online, could you comment on its veracity?
"Narrated By Abu Huraira : I heard the Prophet saying. "Five practices are characteristics of the Fitra: circumcision, shaving the pubic hair, cutting the moustaches short, clipping the nails, and depilating the hair of the armpits." Sahih al-Bukhari, The Book of Dress, Hadith Number:5952"
Elsewhere I've read that there's been fatwas that if a husband requests, the pubic hair must be shaved at least once every 40 days to reduce the smell.