Boots or sneakers.
Sports bra
You can wear casual clothes with a horse riding helmet.
The next best thing to horse riding specific boots are smooth soled work boots with a low heel. You could also wear 'fashion' style boots so long as they have a 1.5 to 2 inch heel on them. If you can only wear tennis or athletic shoes ask if the person has toe cages or Tapaderos to put on the stirrup irons to keep your foot from going through the iron.
Close toed shoes with an inch to an inch and a half heel.
It depends on the discipline. In western you should always wear smooth bottom boots so your foot can easily slide out of the stirrup in case of an emergency.
shoes
you dont wear the stirrup, you put your foot in it!
If the rider is riding the horse they wear a bridle and a saddle and if they are outside they wear a flymask and / or a fly sheet or nothing and if they're in the stall they will wear nothing or a horse blanket.
Shoes!
It depends on what time of year your cattle drive is in and where you are at. If it is warm and dry wear cowboy boots or ropers. If it is cold and wet or just cold wear insulated riding boots. I assume you are riding a horse as well.
Riding boots! These are specifically designed for riding horses and are safe, as they have a heel which prevents your foot becoming stuck in the stirrup. However, paddock boots can also be worn, as well as wellington boots if you cannot find riding boots or are trying riding for the first time. NEVER wear trainers or flat soled shoes as these have no heels and your foot could slip through the stirrups and your foot could become stuck which is very dangerous.
7 out of 8 do not wear a helmet in the USA.