I think you would be okay using raw seeds or nuts (ie sunflower seeds or almonds) if you strain them after making the milk, but you definitely do need to do the cooking step that hespedal mentioned.  This is not just to kill the unwanted bacteria that might sneak in, but if you want the yogurt texture, you will have to thicken it and that requires heating.  The incubation temp (I use my dehydrator at 115) isn't sufficient to either kill off the nasties or thicken it.

As for using soy yogurt as a starter, that should work out fine, but most of the cultures used in soy yogurt aren't vegan to start with.  This may or may not bother you since there probably isn't any milk left by the time it gets to the yogurt, but that is what I've read anyway.  That's why I finally broke down and bought the bottle of vegan cultures off the internet, the couple of varieties that truly are vegan aren't available in my area.

When I made coconut yogurt, I used canned lite milk.  The lite variety worked just fine and it tasted great.  My coconut yogurt was before So Delicious came out with their coconut products, but I bet their stuff would work great as well.  It might be interesting to combine raw nut milk with a bottle of their coconut milk or creamer, or even a bottle of kefir after the batch cools to the proper temperature....

Sounds like another interesting experiment in my future...

I used to make dairy yogurt quite a lot, and I always made it with skim milk, so fat is not required.  I've read a little about making soy yogurt - I guess the yogurt will thicken a little on its own, but it will be more liquid so most people would want to use a thickener unless you're going to use it for drinks.  I have yet to try it, though.  Nut cheese (call my immature, but that does sound a bit funny..) is intriguing.  I think I'll have to try some experiments soon.  I was having a lot of success with the dairy yogurt, but haven't made it since I decided to reduce how much dairy I eat.

You don't actually want the oven on.  If the yogurt gets too hot, you will kill the bacteria.  Ideally, you want it around 100 degrees F.  Our oven only goes down to 170 degrees F, which would certainly kill the bacteria.  Really you can make it in a thermos or anything that will retain heat a bit - you don't necessarily need to actively add heat.  If you leave it in the oven, the oven should be off, but the light should be on (this will give a small amount of heat).  Keeping the yogurt at the proper temperature will let the process go faster, but if your container is a bit cooler it might just take a while longer.  It might be hard to tell with non-dairy yogurt when it is "done," though.  You might have to taste a bit to see if it is sour enough for your liking.

When I made dairy yogurt, the most reliable method was with a cooler and a heating pad as shown here: Yogurt making from Ceres and Bacchus blog.  That works incredibly well.

I haven't tried it with soy yogurt, but I have made dairy yogurt in a slow cooker.  Directions can be found here: Slow cooker yogurt.

The link krs posted also lists a lot of options for incubating your yogurt.

When I made dairy yogurt, I used clean glass jars and also boiled them in water for about 20 minutes.  It turns out you're not supposed to boil the lids with the rubber seal - you could add those at the very end, when you remove the pot from the heat.  If you have hard water, use a little vinegar in the water to prevent a cloudy mineral film from accumulating on your jars.  For plastic utensils that I didn't think I should boil, I stuck them in a pot of boiling water and removed it from the heat.  You can also pour boiling water into whatever container you want to use and let it cool for 10-20 minutes.  I'm not sure if just a run through the dishwasher would be sufficient - I would at least pour boiling water into the containers and over your utensil just to be sure.

I have eaten many many batches of homemade dairy yogurt and never got sick.  The key is to keep everything clean, and work quickly when you are pouring the milk into the containers so they aren't open for very long (so nothing has a chance to fall in).  If you don't eat a lot of yogurt, use small containers - perhaps single serving.  The fewer times you open up a container, the less chance there is of contamination.  If your yogurt has any discoloration (particularly green, red, or black patches) or an off odor, toss it and start over with fresh starter.

So, I made my first batch of soy yogurt today and did a really dumb thing.  I wanted to use tapioca flour as a thickener, but I didn't realize it should be added to the cold soy milk at the beginning of the process.  I added it to the hot soy milk, and ended up with big clumps of tapioca that would not dissolve.  Kind of like bubble tea, but with big, irregularly shaped "bubbles."  Luckily, I was able to strain most of the tapioca clumps out, but I'm not sure if any actually ended up dissolving in my soy milk.  We shall see.  Anyway, I hope others can learn from my mistake!

I feel like I'm talking to myself here, but here goes..  :>  After a couple successful batches of soy yogurt, I decided to try my hand at some coconut-almond yogurt.  It is delicious!  I should probably call it dessert rather than thinking of it as yogurt, since I did use full fat coconut milk.  ;)

I used 1 16-oz can of coconut milk, and made 3 cups of almond milk from scratch (1 cup soaked almonds:3 cups water, strained through fine mesh).  I basically followed the instructions from Bryanna's Vegan Feast (also linked by krs above), adjusting for the smaller batch.  I can't really taste the almond milk at all - even before adding the culture, the mixture tasted more coconutty, but I think the tartness of the yogurt brings out the coconut flavor even more.  I'm not sure if the yogurt would set up much on its own - I used agar and tapioca flour as added thickeners.  I added a little bit of sugar, as with the soy yogurt.