Unfortunately, the answer depends completely on how stringent you are with "Hamilton's rule". If you just mean the equation $r \geq c/b$ then it is important to look at modern usages. In modern usage, all three of the terms $r$, $c$, and $b$ can be arbitrarily complicated. My favorite examples include when $r$ takes into account spatial structure saying that an agent near you is more "related" than a genetically identical agent far from you. Assumptions like that are a must, if you want to look at things that much of modern research is concerned with.
For this "most general" interpretation of Hamilton's rule, only one property is required and this property does not appear on your list: linearity. Note that this is not the same thing as the additivity you describe. Linearity means that your equations for small changes in proportions are linear in fitness: i.e. $\dot{p} = pf(E)$ where $E$ is some description of the environment, and $f$ is a linear function of the variables of interest (I used a continuous or large-population limit here, but similar things hold true for distributions over discrete populations).
Note, that in the above general and super-vague use of Hamilton's rule, the rule itself is not a biological observation but a simple fact of linear algebra. Of course, if you want to get linearity out as a consequence of some biological assumption, then the most common way to do this is by assuming weak selection.
If you put restrictions on what kind of features of the population, interaction, and environment structure $r$, $c$, and $b$ are allowed to incorporate then this will result in biological theories which will be mathematically true under only some of the assumptions you list. The subset of required assumptions will depend on what specific restrictions you place on $r$, $b$, and $c$.
There are no standard assumptions accepted on these parameters, so (unsurprisingly) I can't answer your question further without more specification on your part.
