Denver Health *usually* only hires new graduate paramedics from the Denver Health paramedic school. For outside paramedic applicants I think they are asking that you have at least 1 year experience as a medic in a busy/semi-busy 911 system. It never hurts to apply and interview as soon as possible. Even if you don't get offered a position right away, you will have opportunity to get your name in the mix and get some recognition in future interviews. Don't be surprised if it ends up taking a few rounds of interviews to finally get a job offer, it's not uncommon.
I'm sure you have done your own research but here is a link to the employment page on the division website
http://www.denverhealthparamedics.org/employment
It never hurts to reach out and ask for additional information regarding the interview process before submitting your application. Also, remember you must obtain your Colorado state paramedic certification which can take some time (
https://www.colorado.gov/pacific/cdphe/ems-provider-certification ).
As far as tips for doing well, I think standard interview practice is the most important thing. Practice presenting yourself well, speaking clearly, being concise with your answers, engaging with your interviewers, and generally being likable. You should know or have a good basic understanding of the local protocols (
http://www.denverhealthparamedics.org/images/documents/DHPD-March-2016-Protocols.pdf ), be decisive in your answers to scenarios, and try to be humble/not overly confident. If your end goal is go the fire department route then keep that info to yourself and do not share it during the interview, IMO Denver Health does not like being used as a stepping stone to get on with a FD.
It has been almost 3 years since I interviewed but I'd imagine it's the same process now. It was a general employment interview with a panel of 4 or 5 command staff members of varying rank, and then a second round with medical scenarios in front of another 2 or 3 other command staff members. I think they do some form of phone interviews before inviting out of state applicants but I'm not sure.