Don't worry about the white mold/fungus! It will go away by itself.
AS for snails I would do Rams Horn and Malaysian trumpet snails (MTS). The rams horn will do wonders for the white junk, and will eat algae on the glass, wood and plants, and they won't hurt good living leaves on the plants. They will go after leaves that are dying, so they help with tank maintenance. The MTS stay in the substrate and will keep it aerated and stir up the top 3/4". They will reproduce quickly depending on the amount of extra food in the tank. BUT I found if your feed your fish/shrimp the correct amount the MTS will tend to keep their numbers in check!
Pond and bladder snails have a BAD rep for overpopulating the tanks they are kept in, so use them at your own risk! I try to keep them out of my tanks for that reason! People also dislike MTS for the same reason, but like I said before, I have had good luck with controlling them with the amount of food you feed! Since this is going to be a shrimp tank you will be watching the amount of food anyway, so I would throw in some Rams horn and a few MTS and let them clean the tank for you!
When you can add the snails is pretty much up to you, but I have added them in my tanks anytime during the cycle and I have not ever had a problem. You might want to wait till the ammonia isn't very high to give them the best chance.
As for the Nerite snails I have never used them in Freshwater! They will live in freshwater ok, but they will lay small white eggs all over the tank, and the eggs won't hatch in freshwater, so you have to remove them by hand! To much of a pain for me!
For the wood sinking monapy usually sinks pretty fast, or at least it has for me! If you are going to leave it in the bucket, just make sure to have a rock or something on top of it so it is TOTALLY UNDERWATER! If not it will take longer! Boiling it will help it to sink faster and will also help leach out extra tannins faster, but it is hard to boil sometimes due to the size of the pieces and not having a big enough pot! I use my Turkey Fryer/propane cooker to boil mine, and it has worked very well! Then after I am done I soak it in some bleach water to make sure the fry pot is ready to cook in later on! I don't want a wooden taste in my fried turkeys!!!
You can also do as RedseaReefer said above and pour boiling water over the wood in the bath tub, then fill the bathtub up enough to cover the wood and let it get soaked in it! It is easier to drain off the tannin stained water and replace with clean water in the tub! Just MAKE SURE there is NO KIND OF TUB CLEANER OR SOAP in the tub before starting!
Lastly on the whole cycle thing. If you are not keeping a food source for the bacteria to continue to grow while you are waiting to add the shrimp, you truly not going to cycle the tank completely. Really what I am trying to say is that you need to add some kind of food for the bacteria to eat and grow yourself a good colony! The little amount of bio film that will develop on the plants, wood, glass, and substrate won't be much with out something living in the tank or adding ammonia to "fishless cycle" the aquarium! I have NEVER done the "fishless cycle" myself! I at least start with a few hardy fish and snails. Since this is going to be a shrimp only tank I would only add snails! Then feed them just a small bit of food ever day to every other day, till your ammonia readings peak then fall, then your nitrites will do the same, and finally the nitrates will rise when the nitrites start to fall. When your are reading 0 ammonia, 0 nitrites, usually after about 4 weeks you will be shrimp ready.
You may have already have know all of this, but I didn't want to see you add your beautiful shrimp and have them suffer through any kind of ammonia spike!
I hope this helps ya out,