Physics in concerned with space and time as perceived by observers. Space and time are ways for an observer used to precisely discribe what he sees. You have a clock, you can see things around you and measure angles between the rays of light (or other things) that hit you coming from what you look at. Physics does two things:
It restricts the possible things you might see later given what you've been seeing until now. In optimal cases it even predicts exactly what you will see or probabilities for the possible things you might see.
It tells you what other observers see when looking at the same thing as you, given where they are (where you see them) and how they move relative to you (how you see them move).
Anything besides this (spacetime, electric fields, temperature etc.) is just pictures helping your intuition guess the results above approximately, given approximate input. It is a common misconception that interpreting these structures has physical meaning. Don't get me wrong, intuition is extremely important both for using theories of physics and especialy for creating new ones. However you are asking "am I thinking the right way as far as Physics is concerned" when your thinking does nothing to help intuition with predicting the output of a theory built around spacetime. (If it does, I don't see it and you should explain. Find physical situations when your way of thinking suggests a wa, of guessinv what happens.)
As such, your question can't be answered. The analogies you provide may or may not be "good" for math, which is for the math forum to decide. Physics has nothing to say to that. (That is my opinion, which is the only thing you can get as an answer to that question, as opposed to facts.)