Hi and welcome!
I am in Edmonton, and had a hospital birth here in 2006, and a homebirth here in 2008. To answer your questions....
1. Yes, sadly it is very difficult to secure a homebirth midwife here in Edmonton. My most important advice to you is to interview and select your midwife now, BEFORE you get pregnant. There will be no time to make such decisions once you get a positive pregnancy test. In the entrance/exits of most major grocery stores, you can find the racks with the free local newspapers/magazines. Pick up the one called Birth Issues. In there is a list of all the midwives in the Edmonton area. Actually you could probably check their website too, google Birth Issues and/or ASAC (Association for Safe Alternatives in Childbirth). Start phoning and booking consultations with midwives so you can find one you are comfortable with, then decide who your 1st choice is. As far as I know, most of the midwives practising in Edmonton are Registered Midwives, but there are some who are Traditional Birth Attendants. My midwife was an RM, and I truly don't know much about TBAs.
2. When you get your positive pregnancy test, phone your chosen midwife immediately to tell her you are pregnant and want her care. Do not wait. There is no need to see a doctor to confirm your pregnancy. Your midwife may not need to see you right away, but she will need your dates to reserve your spot with her. Some midwives hold spots for previous clients. Some midwives prefer to take homebirth clients over hospital birth clients. She should be able to tell you right away if she can take you on or not.
3. I can't speak for all midwives in Edmonton, but the one I had did all my prenatal care from the moment I got a positive pregnancy test, until my baby was 6 weeks old. She gave me a requisition for an ultrasound, and lab tests like the GD screening, she also did the GBS swab. She did all my baby's newborn screening & tests and Vit K shot. She secured me a referral to a breastfeeding clinic when I was having trouble with BFing in the first week or two postpartum. She did all my homecare visits postpartum. She provided all the necessary paperwork for registering the birth. There was absolutely nothing that I needed to see a doctor for. Now obviously if there is a complication that arises that is beyond her scope of practice, then she will refer you to the appropriate specialist.
4. As for a natural birth in a hospital, I will say it's possible but not easy. Some hospitals are better, some worse. The most natural-birth-friendly hospital program in Stony Plain has sadly now closed. Some midwives will do hospital births, but they're rare, and most midwives that I've heard of will preferentially take homebirth clients. Certainly some doctors are more NCB friendly than others. If you have a doula, that will surely be a benefit to you in a hospital birth. Some of the busier hospitals don't have a lot of patience to wait for natural birthing moms. Some hospitals will advertise that they have birthing tubs available - but then you find out that the tub is not allowed after your water breaks. Honestly, the easiest way to have a natural birth in a hospital might be to stay home during labour and show up at the hospital pushing!