If I was in charge of F1, I'd cut the championship down to 16 races, as many (or as few) teams as there is, albeit with each having an equal number of cars (whether that number is 2 or 3 I don't care).
I'd have 2 compounds of dry tyres (hard & soft), with the harder tyres lapping the circuit slower than the softs, but with less degradation. Degredation would be less extreme than it is now, so for example on average, a set of softs would last a third of a race, whereas the hards would last half the race (thus reducing pitstops to roughly 1 or 2 a race, although the individual track surfaces would throw a small bit of anomalies here, but not too much). And I would remove the 'cliff', to allow the driver to push more at 100% for far longer on the tyres.
I'd also remove the fuel flow limit, but ensure that each car is filled up to a minimum level (to prevent under-fuelling so drivers can push at 100% the majority of the time without having to resort to fuel-saving modes).
I'd tweak the aero to allow a driver to catch & pass one-another without the need for devices such as DRS, and i'd make any push-to-pass ERS systems deployable at any point around the circuit (I think this is how it is already though).
As for tracks, it would be set on a country basis, not a track basis, and then i'd allocate a set of 'core slots', that must have a grand prix on the calendar:
1. UK
2. Monaco
2. Italy
3. France
4. Germany
5. United States
6. South Africa or Brazil (Can't make up my mind)
7. Middle Eastern Country
8. 2nd European country (Designated as European Grand Prix)
The other 8 rounds would be decided 2 or 3 years in advance at an annual ceremony (The country's national motosrport authority would put in a bid for a particular track- with a restriction on no more than 2 bids from 1 particular continent). And then the bids are put into a pot & drawn at random to get the 8 rounds - they then all get together & decide the order for the season.
The reason i say the 1st 8 'core' rounds should be decided on a country rather than track basis, is because I don't think the average fan cares say whether the British Grand Prix is at Silverstone or Brands Hatch, or whether the Italian GP is at Monza or San Marino. Circuit hosting contracts would be (obviously) for the awarded year in question for the 2nd 8 rounds, and no more than 3 years in a row for the 'core' rounds. If a 'core' circuit has just lost it's contract to another track in the same country, they have first opportunity to submit a bid in for their country in one of the 2nd 8 rounds (if they're still desperate to remain on the calendar).
Team prize money etc. should be distributed equally, with a constructor's championship bonus directly in proportion to their championship result (set so that even the last-placed team gets something). Finally, the bottom 3 placed teams should get a bonus (something like 30mil for 12th, 20mil for 11th and 10mil for 10th). 'Heritage' payments to long-standing teams like Ferrari, McLaren etc. would be abolished.
Regulations (both technical & sporting) would be devised through a panel of former team principles/technical directors, whether from F1 or otherwise (they must have zero links to any existing F1 team or team owner), along with by representatives from the FIA. Regulations must be devised & approved no later than 2 years before the season in question, ideally 3.