We love to see photos of the property in question - that helps with our questions for you. The forum doesn't let very large (only about 250k) photos up, but if you can post small size or host from flickr or somewhere, please add some to this thread.
The first year of any garden is often the most difficult because there are old seeds in the soil, it is compacted, you need to figure out the drainage, the spacing, how to get the water to it, etc. And in the organic program bed preparation is the most important part of this. Visit the Organic Guides
page and follow some of those links, starting with the basic gardening and bed preparation information. You'll find yourself bringing home bags of rock powders like lava sand and greensand and decomposed granite to mix into the soil and not going near the chemical weed and feed or other toxic products.
Compost doesn't need to be hot to break down, though I have friends who are hotrod composters who aren't content until they can see steam coming off of the heap in the morning and measure high temperatures with their fancy long thermometers. I have a large yard and every calendar year I start a new pile. I turn it over every so often when it's small, but as it gets to the end of the year I just pile it on, usually concluding the year with all of the dead plants from the summer's garden. Last year's compost is just a dormant pile in the yard, and the one I used in the garden this year was from two years ago. Over that much time it becomes compost even without discernible heat. You can build structures, you can get bins that turn, you can drill holes in 55 gallon barrels and roll them around - look in the forums here and there are lots of questions and answers about compost. And yes, you can put dog or cat waste into the compost, it isn't a problem. Only put meat type food waste in if you aren't troubled with animals getting into the compost.
As to your specific remarks, I wouldn't till the compost ingredients into the soil if they aren't already composted, as you say you did, but let them sit, water it if it gets dry, make sure there is some green stuff (grass, weeds) or leaves in it and turn it over every so often if you want to see some heat.
I wouldn't worry about a weed barrier, but before you get set to put in more raised beds you might want to pile all of your mulched leaves on the spot and let them break down. Usually a thick layer of leaves will kill weeds and grass under them, and then you can leave them in place or mix them in with the soil you put into the raised bed. I simply pile up and bevel the sides of my beds, I don't build a frame around them, and by the end of summer the weeds are pretty well established in the paths, but the garden is taller and grows above them. Mine isn't anything like Martha Stewart's garden, but it still produces well.
Once you have your beds established, you need to be sure that the biological activity in the soil is healthy and robust. Use things like beneficial nematodes before you plant to avoid problems such as cutworms mowing down your bedding plants and other pests through the growing season. Using something like Thrive
(Alpha BIO systems) to build up the mycorrhizal fungi that helps keep the soil healthy and the roots happy. Instead of chemical fertilizers, there are sugars like dry molasses and by-products like corn gluten meal and various
There are lots of organic remedies for pests and for soil deficiencies, and there are types of soil tests that can tell you what in your soil is available to the plants, if you want to send out soil for a test.
Beyond this, it's a good idea to see what your neighbors are doing. Since you're asking on an organic gardening site, the first question you need to ask area gardeners is if they are organic, and where do they get their products and advice. And decide if it will work for you, or if you need to do more research. This site is based in North Texas but there are are organic gardeners all across the US and beyond who rely on this site for useful basic information, information about new research, and troubleshooting garden problems.
If you are starting your own seeds, then you need to figure when is the last frost in your area (what is your USDA zone
?) and then give yourself a few weeks prior to that for the plants to get started.
Good luck, and please post photos!