This can be effectively addressed by strategically placing the mini-split units in locations where they can facilitate airflow into adjacent spaces, and by implementing simple modifications like undercutting doors to promote air circulation.

Very efficient.Chances are the older system has substantial duct loss and should be tested.Mini splits have their place but cold air is heavier and drops so the bottom floor doesn't make sense.Electric motors do not raise in load consumption as they age, it's the mechanical work done that raises load, typically deferred maintenance on belts, shives and pulleys.I'd disqualify HVAC companies by asking; Are you proficient in the new Manual J,S & D standards (that will be required in the industry moving forward)These companies will understand laminar flow, stratification and static airflow imbalances that have a great effect on efficiency, operation and longevity of a conditioning system.Get 3 estimates, compare scope & make sure its permitted.

Is there an actual reasonable fix to this or do I have to install the fancy heat sensing flaps that will close off air flow going to the bedrooms???

It would also allow for air flow up there too.I'm looking for anyone who has experience and thinks it's a good idea, or anyone who can think of a reason not to do it.

My HVAC hadn’t no air flow in my house for weeks.

But it's hard if you don't have good airflow between rooms.

There's no real 'fair' way to truly separately meter the natural gas used for the different units, unless you get into rabbit hole land of re-inventing a wheel and a series of sensors measuring heat and air flow within separate ducts, on your (customer) side.I'll comment on each option individually and your decision will be numbers driven. 

We're trying to decide on what material to use to replace it.There's not great air flow to inside which is why we're considering moving away from carpet and something that may hold less moisture, easier to clean, look more modern etc.

Air Flow - There is a window in the room and I have been advised that an AC unit can be added to this to keep the room cool2.

The small mesh size does limit air flow through the screens.