Get a block of paraffin (available at most grocery stores).
Melt the paraffin in a can or pan. Place some lint in the sections of the egg carton. Pour paraffin in the section to cover the lint.
When the paraffin sets, cut the egg carton sections into single units.
When you need a quick, sure fire, just use any flame source (I recommend the wand style gas lighters) and ignite the paraffin and voila, hard to extinguish fire.
I have started wood fires without tinder with 100% luck. No smoke, no smell except the wood you choose! They burn for 10+ minutes.
Good luck!
My filler/wick material of choice is small wood chips. The little stuff that's all over the place after splitting a pile of wood for the wood stove at home works. Break the longer bits into pieces 1" or shorter.
Paper shreddings also work well for the filler/wick material.
The rest of this looks good.
If you're in an cold climate area, a "heater/stove" is a good idea. A sterno type works, though you can make your own with a metal coffee can, a spare roll of toilet paper and a bottle of rubbing alcohol. Have a ceramic, or stoneware plate, or a metal cover to snuff it out. You make the "heater" by removing the cardboard roll from the center of the toilet paper, placing it in the metal coffee can, and filling the can with alcohol (rubbing alcohol is fine, but if you can find 90% or 95% rather than 70%, it will be better). Pull a "Wick" up from the center of the roll and light it. This can be used as a heater, or as a makeshift cook stove.
Use the saw to cut a small baton length, and cut the rest into useable lengths. Use the baton and your knife to split the rest of the wood for the fire.
Another approach is:
1. Get a wooden dowel about 5/8” in diameter and wrap a sheet of newspaper around it, and then tie a piece of string around each end and the center (or put on three rubber-bands that aren’t very snug).
2. Remove the dowel and staple one end of the paper tube securely closed. If you used rubber-bands in Step 1 and they collapse the tube, they’re too tight.
3. Prop it up vertical, fill with melted paraffin, and let cool/solidify. You don’t want to be spilling the melted paraffin on your hand.
4. When cool/solid, cut into pieces roughly 6” to 8” long.
You’ve now got what are essentially candles with exterior wicks.
You can light the paper, use it to light a fire (as the paper burns, the wax will melt and can be poured-out to assist in lighting the fire), extinguish, let cool, and stick it back in your pocket.
Multiple times.
That idea's a keeper...