For the original PDE, the positivity can be deduced from the maximum principle for a parabolic operator. There is also a discrete version of the maximum principle for the finite difference parabolic operator as for example stated in this paper.
Some of the standard tricks are mentioned in this paper, Finite Difference Schemes with Exact Recovery of Vanilla Option Prices
which also shows how to set up the finite difference scheme so that all vanillas with strikes and expiries on the grid are matched exactly.
Don't solve the Black-Scholes PDE, solve the heat equation
One of the major results of mathematical finance is showing that the Black-Scholes PDE can be mapped to the heat equation. The heat equation is both mathematically nicer to handle, analyse, and computationally has much better solvers than other generic PDE solvers. Don't solve the Black-Scholes PDE, ...