They are a lot of open problems in market microstructure. To have an idea of the whole landscape, have a look at Market Microstructure in Practice, 2nd Edition, by L and Laruelle.
I would split them in
From the viewpoint of exchanges
Optimal fee schedules to "attract" liquidity (and hence efficient market makers), have a look at Optimal make-take ...
There is a cascade of methods to choose a settlement price in futures markets - starting with trades in the relevant market, and going through trades in other expiries (plus spreads), quotes, quotes in spread markets, trades in related markets, previous day’s settlement prices etc.
This answer to a related question may be helpful - https://quant....
"How to roughly measure how much premium investors would demand if a stock could not be sold and its investors had to stick with it permanently using just dividends not capital gain as return?"
A starting point might be to look at the private/public valuation arbitrage in the sector.
The usual measures are trade volumes (from all sources of trades / order books), bid/offer spread, order book depth, quote ages, trade frequency, etc. For quick comparisons, average daily volume is the best (and easy to obtain).
I will attempt to elaborate on this from risk management perspective.
scenario analysis approach: An example of this is stress testing that Fed mandates for investment banks. Fed gives stress variables to various fundamental macro variables. For example, a certain market stress scenario will be rates down 100bps, volatility up 30%, curve flatter by 30bps, ...