Can I get away with grass seed???

I am in the process of removing moss from my lawn and it is looking like I am gonna have a couple of patches that are about 2sqm each that dont look like they are gonna have much grass left on them at all as well as at the bottom of the garden that is about 10sqm that hasnt got any grass on it at all, the bottom of the garden is shaded through out most of the day unfortunatly. I think most of the damage was done a couple of years ago when the garden was completly flooded for 2 or 3 weeks before I bought the property.
Just to complicate things a bit more I have 3 boys all under 10 and a labrador so its gonna be just about an impossible task to stop the garden being totally used for a long period of time.
I am hoping that you think I can possibly get away with killing and removing the moss, killing off any weeds and then reseeding the entire lawn obviously with more emphasis on the bald patches.
Can I get away with this?? In an ideal world I would have it all re-turfed but I hope to have the garden fully redesigned next year but cant afford that just yet, so really need to do a half decent bodge job for this year so the kids can go out and play on grass instead of on soil as the summer progresses.
If I can get any tips or advice they would be greatfully recieved, even if its just what seeds,fertiliser, lawn food etc. to use and also how long will it take before I can see some grass on my soil do you think??
Thanks in advance

 

Basically, yes. You can re-seed. Mix seed with a fine compost or top-soil and broadcast over the areas to be done. The patches are probably best dealt with by raking the surface first. Seed will take about 10-14 days to germinate. I would try and avoid walking on the seeded patches for a month.