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