Starting seeds is a tricky business (not) - ha ha. The seeds WANT to germinate, it's in their genes. You need to provide the right conditions: usually moisture and a moderate warmth. Some seeds like it cool (ie, spinach) but in general, all my plants start great on the heating pad. It took 3 days to get lettuce germinated in 4inch pots this week.
I use a sterile potting soil for starting seeds. I find that watering the mix when planting compacts it enough, no need to pack it before putting seeds in. Do of course press the seed into the mix for good seed/soil contact, but again, no need to compress significantly. If you don't have a source of heat, you can use hot - but not boiling - water to wet the mix before seeding. You should put a thermometer in your cupboard - if the temp is 70F or under, your're fine. Just remember to check your seeds every day!
You may have to work a little to find the right timing - some plants started too soon will get leggy, esp. lettuce. I've never had problems with tomatoes started in Feb/March (Zone 5) b/c by the time the plant wants to really start growing, I'm setting them out during the day for at least a few hours, and they get pretty stocky.
The looseness of your starting mix is a good thing, because plants need air and the developing roots will find it no burden to stretch out and grow! Which leads to the care and feeding of the young plant
- keep an eye on your seedlings, they can suck it up at stages of development. Water when needed, but don't drown them! I like to "pot on" some plants - tomatoes, broccoli, lettuce, spinach. Seeds of these plants started en masse in 4in pots can be transplanted when you see the first true leaves on the plants.
After you get seedlings, they will want light - as much as you can give them. I use plain 'ol flourescent shop lights. Set the plants right up to the lights, unless you are lucky enough to have big window. I visit seedlings daily, to make sure they are not drying out, and to rotate the plants around the lights (dimmer at the ends). After all this, I make sure to harden the plants off before setting them outside. Even if you are going from GH to outdoors, you should take the 3-4 extra days to harden the plants off and get them used to the rough world/full sun.
Can't speak to the compost question, as I never have enough to bring inside. Remember that even plants that are not recommended to transplant can do just fine if handled with care. For example, I start my sweet corn indoors and transplant - the only problems I've ever had is when I used peat pots (b/c you can plant the whole pot). Big mistake, I won't do that again! And chinese cabbage never seems to give me problems when I TP it :shrug:
Most of this info is already out there - you can google seed starting or starting seed and have a couple lifetimes of reading to do :)
Good luck to ya, and don't forget to go through your seed packets BEFORE you order new ones :D