There's a lot of good stuff in this thread already. Here's my 2 cents.
Where I'm coming from: I'm a backpacker. When the power goes out, I view it as camping in a large, wooden hard side tent where I don't have to move things to the woods. I merely break out my backpacking and / or car camping equipment to cook / light with. I eat as I would when in the field, which is easy since I usually draw my backpacking food from the pantry.
In your particular situation, here's what I'd suggest:
1) At the next opportunity, take the generator in to have it serviced at your local dealer / small engine repair shop, or call a more mechanically inclined friend. Get them to fix it, then tell you what was wrong. Take care of that next time yourself. Pay off friend (if that's the way you go) with home baked goods / nice home cooked meal / 6 pack of beer. Be sure to keep 20 gallons of gas on hand (generators are useless without fuel). Use fuel stabilizer in the gas cans and rotate the fuel every year or so - pour into the car, then refill can in lieu of the 5 gallons you just poured into the tank. Use sticky notes / masking tape labels so you know when a gas can was filled and can keep track of the rotation.
2) I'll second the Coleman type propane stove*. Attach to one of the hoses from (3) below.
3) For the stove and lantern (below), get yourself a "post" and hose set up that fits onto a 5 gallon / 20 lb refillable bulk cylinder you likely already have for your grill. 1-2 hoses, one post. Google "propane tank post". This type of equipment can be found at a Cabellas / Big 5 / Wallmart. I have the Coleman 30 inch distribution tree.
4) Get yourself a 2 mantle propane lantern, also at Cabellas / Big 5 / Wallmart. Attach to the top of the post from (3). This is both light (~200 watts incandescent equivalent) and a fair amount of heat.
5) You can get an indoor safe propane heater. Google "indoor propane heater". Use one of the hoses from (3) to attach this to the bulk cylinder.
6) In re the water situation. Check out REI / MEC / EMS websites. Go to their backpacking equipment section. Go to the water filter / purifier section. Look for one of the larger gravity type filters (bag to bag, with a filter in the line between the two). You mentioned you have a pond you drew water from last year. Is this pond totally gross (as in full of deer / duck shit and farm residues like pesticides or fertilizer or totally scummed over) - if not, then just filter water from the pond using the filter system you choose from REI / MEC / EMS. You'll have an unlimited supply of drinking water. The filter will take care of bacteria. If you're worried about virus' (generally not a problem in North America), then 3 drops of typical bleach per gallon after filtering, mix and let it sit for an hour or so. I have a creek ~1/2 mile from the house - if needed, I'd go there and draw all the "raw" water I needed and treat it at home using my standard backpacking methods.
The full set up for 2, 3 and 4 can be had for under $150. 6 - depends on what you choose, but I saw a fine system at REI for under $100. 5 is a luxury...although with small children, keeping the house above 50 is important. You may also want to buy a 2nd 20 lb / 5 gallon bulk propane cylinder. Use the one from the grill until it's empty, swap in the fresh full one, then go and fill the other one right away in typical usage. That way you always have at least 20 lbs on hand, which should be enough for several days of heavy use, including the heater.
* - I led a volunteer trail crew into the Wilderness earlier this summer. We ate like kings and queens using nothing more than a 2 burner propane stove and a cast iron Dutch Oven (DO for only some of the meals - others were 100% on the stove). Lasagna, stew, fresh biscuits and dinner rolls, bacon / eggs, pancakes, etc - granted, the Lasagna was in the DO, but we could just as easily done spaghetti with canned / jarred sauce on the stove. Our crew was 9, and we had a couple Forest Service folks with us for a few days as well - so feeding a family of 7 would be quite doable on a 2 burner stove. We fed up to 11 adults doing hard physical labor. We also had a couple gravity bag to bag type filters that easily supplied all our water for both cooking and drinking.