Re. question 1, the preceding assumption is relevant since it makes clear the question is about the relation of map to territory, (private space to physical space).
"Assuming that there is physical space, and that it does thus correspond to private spaces, what can we know about it? We can know only what is required in order to secure the correspondence."
So I will complete the answer with a quote from Gregory Bateson:
What gets onto the map, in fact, is difference, be it a difference in
altitude, a difference in vegetation, a difference in population
structure, difference in surface, or whatever. Differences are the
things that get onto a map.
A difference is a very peculiar and obscure concept. It is certainly
not a thing or an event. This piece of paper is different than the
wood of this lectern. There are many differences between them - of
colour, texture, shape, etc... Of this infinitude, we select a very
limited number which become information. In fact, what we mean by
information - the elementary unit of information - is a difference
which makes a difference. - (Steps to an Ecology of Mind, pp.457-459).
source: web archive
The differences are what is required to secure the correspondence.
Re. question 2: "What does 'properties of relations' mean?" Properties like altitude, a difference in vegetation, etc.
Re. question 3: "Lightning and thunder, really happen simultaneously, but we perceive one after the other. Isn’t that a change in order?"
Presumably Russell is saying that in the private space lightning and thunder is known to happen simultaneously, as they actually do in physical space.
Hence "the time-order which events seem to have is, so far as we can see, the same as the time-order which they do have."
He also says "it must not be supposed that the various [phenomena] have the same time-order as the sense-data which constitute the perceptions of those [phenomena]."
So you hear lightning and thunder separately, you are aware they happen simultaneously, and in physical space they do happen simultaneously.