Ash: You are on the right track judging from your posts so far.
Regarding 'proper distance' and other cosmological measures.
It's useful to keep in mind that the the best fit cosmological model (the 'standard' one everybody uses) is called Lambda CDM [LCDM] cosmological model. If you use this model, and that's the standard in these forums, then 'distances' and 'times' can be compared with others without the ambiguity of unconstrained general relativity by using the specific assumptions in the model.
The LCDM model is the fine-tuned version of the general FLRW [Friedman...et al ] where the parameters are chosen to get the best possible fit to our universe—that is, to match observational data.
That standard cosmological model uses some standard assumptions, like homogeneity and isotropy, already mentioned in previous posts, and standard inputs to provide answers which people calculating with it can use to compare results and discuss 'distances' and 'age of the universe' without unnecessary confusion. For example, as noted by Powell, the single frame of reference is agreed upon the be that of the CMB...being at rest at each end of the 'distance' with respect to local cosmic microwave background in both places. Oddly, it seems, when you first hear this, because who measures 'distances' here on earth, for example, with both ends moving relative to each other.
When you read things such as: " The present is year 13.7 billion of the expansion of the universe and we are receiving CMB from hot matter that was 42 million light years from us when emitted ..." you are using the standard conventions of the standard model. If you are not at rest with respect to the CMB, time and distance will not be the same.
Regarding the balloon analogy: just don't make any analogy between the rubber material and empty space...focus on the dots changing distance as the radius expands. Nor can you make an analogy that if you decrease the radius of the balloon to zero everything started from a single point in space. That's not part of the analogy. It's an analogy not a perfect model over all conditions.
And that reminds me, neither is the standard cosmological model a perfect model over all conditions...Assuming homogeneity and isotropy enables solving the Einstein field equations; the model starts after inflationary expansion concludes...so we have two models 'glued' together at the front end. And before that,as far as I know, nobody has a generally agreed upon model for the big bang itself.