@tonynsx Welcome to the RobotShop Community. It’s quite an involved question and requires roughly the same knowledge and expertise as converting a real passenger vehicle to RC control.
I think I need something to turn the steering wheel. {…} I think for turning I can use something like a car windshield wiper {…} motor with a motor controller?
This depends on how many rotations the wheel needs to make. If it can be limited to 270 degrees (plus or minus 135), your best bet is a high torque RC servo like:
If not, you’ll need a gear motor and motor controller. The motor you mentioned has no gearing so it is good torque compared with other non-geared motors.
something for brake
Assuming the motion is linear, consider a linear actuator:
https://www.robotshop.com/en/standard-actuators.html
Ex: https://www.robotshop.com/en/4-stroke-firgelli-automation-12vdc-35lb-force-linear-actuator-w--pot-feedback.html
(You’d need to get an idea of the force and travel needed).
speed levers.
Would need to see what you have.
Also, I like to control this from inside my house.
Having obstructions between the transmitter and receiver are not great, especially when the machine is large and can cause significant damage if something goes wrong. A good RC transmitter should work well, but safety first.
https://www.robotshop.com/en/wireless-rc-pwm-servo-controllers.html
Eventually, I like to add a microcontroller to this project also. I’m familiar with Python, so may be an ESP8266 board will work for it.
It will be important to choose parts which can then later be used with a microcontroller, like the linear actuator with built-in potentiometer.
Most important - safety, safety safety. Have ways to stop the machine dead in its tracks both remotely and by pressing an emergency button on the mower itself. When automating passenger vehicles, there is often a redundant system in place.