Hi Andrew,
I will have a go at answering your questions, even though I am no particular Heuer expert or watchmaker. I do, however, have a lasting interest in Rally clocks.
From your explanation, I am assuming you wish to hold on to your inheritance and nice (but worn) patina for family reasons, rather than maximising their value for re-sale.

If I were in this situation, this is how I would see it. Otherwise, please let me know.
1. Do as little as possible to get the Monte Carlo going at minimum cost.
2. Find yourself a watchmaker, skilled in these pieces. Members on this Forum will gladly tell you where they are, depending upon where you live.
3. Maybe a light service, then leave the Mastertime alone.
5. Regarding the Monte Carlo. Find out which parts are needed. You might need small parts on both pushers, like return springs, etc. ( If you decide to go down this route, I do have available a spare threaded collar, left-hand pusher and movement).

Other questions. Both of these pieces were made/ sold in the late '50s to late 60's, so they are complimentary. Unless you have box(es) and papers (stamped guarantee or invoice), they are unlikely to be a Rally Pair, originally sold as such. Unless the Mastertime has been re-conditioned, the differences in patina seem to suggest they are from different sources.
Mastertime. 'On the Dash, Dashboard Timers' site, you will also find '63' shown on an Abercrombie and Fitch Mastertime. I do not know the relevance of this.
I have never before seen a white-banded second hand.
Monte Carlo. You will find the dial almost impossible to refurbish, although New Old Stock dials and crystals are available on Ebay. I have a 3-button Monte with similar serial numbers, but no 'H', so cannot help with the relevance of this. Similarly, I would not worry about the absence of a marking. This site is full of lovely watches where people have done unexplained things with them. Certainly, it is unlikely to be anything except the genuine article.

Finally, get yourself a nice varnished hardwood 'Toblerone' to mount them and they will always be a talking point for family and friends, displayed on your desk or mantlepiece.
Better still, get yourself a nice MGB to use them in!
Hope this helps. Best Regards.
David. Oxford. UK>