Removing stickers from the door

There are lots of concrete cleaning products on the market, but I have had good luck with a 1 gallon solution of 1/4 bleach, 1 cup TSP and 3/4 water. Assuming you are talking about a driveway or garage floor, I would pressure wash the area first to get off the surface crud, then lay on some of the solution and let it set 5 mins or so. Scrub it down with a stiff broom and power wash again. Repeat this as necessary.
As a last resort, you can also use some muriatic acid, but test a small area first to be sure it doesn't effect the color too much.

Lattice is really cheap stuff. It's probably easier to just get new lattice and rebuild the gate. Or, for a simple fix, get a wire stripper wheel for a power drill and take at it. Get as much off that comes off easily and then cover it all with quality primer.

FYI, if a lot of the wood has been weathered for a while, it's going to be a chore to get primer to stick to it without a lot of sanding. Again, might be easier to replace the lattice al together. You can also usually find synthetic lattice that doesn't require any painting.