I only need you to answer whether US. private businesses set their own prices- and compete among another and, if the answer is no, provide examples to that effect, such as laws.
You are ignoring
[ignoring?? So far, my goal was for you to answer a simple question - which you did below] the indirect influence of the government which is as huge as our planet, and want me to focus on the direct influence of the government on prices, which is still present but tiny?
Alright I give you an example:
Slot machines: In some gambling regulations the slot machine owners are forced to set the payout ratio to at least 0.7:1 ,cannot set any lower, so the slot machine owner casino can set it's machine only between the (0.7,1) boundaries. That is direct price control.
Minimal wage: It is the price of labour. The government sets a minimum price of the labour, and cannot be below it.
Rent control: Price of rent. It usually has a maximum price/ square meter of the residence.
And so on.
There is your price control, but these direct controls are still too tiny, compared to the indirect controls which are huge.
Your three examples, however seemingly plausible, are not really applicable, because they apply to the consumer. Remember that we are discussing
setting price among each other, and not business-consumer relations.
Also, your slot-machine example is not really price control at all, because the transaction here is 'money in exchange for playtime with the machine', where the money-part is not controlled.