How to prove Feller property without using heat kernel estimates
In the case of a diffusion, (1) is implied for example by having bounded coefficients. This follows immediately from applying BDG to $X_t-x$ and doesn't require (2) which is much harder to get. Note by the way that (1) itself is quite a bit overkill since it rules out the OU process, which is the prototypical example of a Feller process, even in your "strong" sense.
I also want to point out that the definition of "Feller property" isn't consistent across the literature. For many authors, mapping $C_b$ (continuous bounded functions) into itself is sufficient for a Markov process to be called "Feller". Mapping $C_\infty$ to itself rules out simple examples like $dx = -x^3\,dt + dW$.
The answer depends on what you are working with.
If you have a pseudo-differential operator and ask whether it generates a Feller semigroup, the best answer known to date is contained in Walter Hoh's works; see a three-volume book Pseudo-Differential Operators and Markov Processes by Niels Jacob for rigorous statements.
If you have ask what differential operators generate Feller semigroups, or what SDEs determine Feller diffusions, then this is closely related to a question about the possibility of starting a process "at infinity", and in some sense dual to the "no explosion in finite time" problem. This can be a delicate question. Feller property is known if the coefficients of the generator grow slowly enough (quadratic growth for the coefficients at the second order derivative and linear growth for the linear term). For a detailed discussion and references, see Properties at infinity of diffusion semigroups and stochastic flows via weak uniform covers by Xue-Mei Li.
For general conditions, you may see a Primer on Feller semigroups and Feller processes. For example, Theorem 1.10 gives an equivalent condition expressed in terms of transition kernels.