That's spider mite damage alright.
I think the only way neem oil can wipe out spider mites is by making them die laughing: spider mites are very tough and neem oil is a very bland insecticide, and I am still being generous.
You have two alternatives to eradicate spider mites: either a mineral oil or a miticide (abamectine to say one). Anything else is just wasting time and money.
Fungus gnats can be eliminated using Bacillus thuringiensis var. kurstaaki (BTK): it's the same stuff that works so well on mosquito larvae. You can buy it either in granules (Mosquito Bits) or powder (Gnatrol): in my experience the strain used by Gnatrol is considerably more aggressive than that used in Mosquito Bits but both work well.
