Regarding charging for funerals, I sing for both weddings and funerals, and I do charge less for funerals (about 25%). I make it clear on my website that this is a discount “due to the nature of the event”: I believe that music has a real purpose in the grieving process, and that everyone should have access to good music at such a time, regardless of income. It’s also because funerals are a *lot* less work. Grieving people are much less choosy than brides, and there is so little notice (usually 2-3 days) that there isn’t time for the client to flake out. If you can get into playing for funerals, it’s a great bread-and-butter gig. Good luck!
(What I specifically charge probably won’t help you much; I live in a very unmusical place, so my rates are quite low.)