I have read some websites that suggests that there was a cap on the bitcoin price of around $1000000000 and caps on the dogecoin price at $15000000. Howether, not sure if these are legit and they have changed in the past!
There is no 'cap' on the price of bitcoin. The price of bitcoin is determined by the market and not by its code.
Yes bitcoins are not controlled by even any big institutions or governments but bitcoin price is purely depends on demand and supply. If demand for bitcoin grow the price will grow.
No, they are not controlled by supply and demand.
Anyone with enough bankroll and statistical approach can change the bitcoin prices by playing with buy and sell order list on any small exchange.