PTO red light switch is on when PTO isn't, so engine won't start

slack belt or electronic clutch issue.

https://www.google.com/search?q=Cub+Cadet+LT1050+PTO

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The wiring diagram shows the reverse switch is normally closed until activated. This would mean forward travel requires switch closed else PTO disengages... same as your experience.

From your description of progressive failure I suspect the switch has failed or has come loose and moved from it's normal position.

Go to the JD website below and search "reverse switch". A parts diagram should display. Look for item 15, part number LVA13307

https://partscatalog.deere.com/jdrc/navigation/equipment/61166

The reverse pedal or linkage must somehow activate the switch.

There could be a problem with the PTO switch or safety interlock module or electrical interconnections so if the reverse switch checks out, further troubleshooting will be necessary. Start at the reverse switch and work your way to the PTO switch then the interlockmodule.

Have you checked the safety switch under the seat? Operator presence is required on most equipment today to properly function. If the seat switch isn't closed, it will shut the engine down if the parking brake is released and/or the blade PTO switch is engaged. Unplug the seat switch and place a jumper into the seat switch receptacle to the wiring harness; not the male plug to the seat switch. If you can start the mower, disengage the parking brake, and engage the mower blades (with your other switches reconnected) without engine shut down, replace the seat switch.

The Reverse Interrupt Override switch is supposed to let you keep the blades engaged while using reverse. Be sure PTO is off and brake pedal is depressed and try to start engine. If it wont start/ wont crank then a switch is stuck or failed or there is a broken wire. Switches that are involved with PTO operation are the seat switch, reverse switch, PTO switch, and RIO switch. The starting system requires neutral ( no reverse activated), brake switch and PTO switch off.

what Model Cub?

All the Safety controls pass through the PTO Switch on many of the older Tractors.

test the safety switch on the brake pedal and the pto switch

This sis a micro switch issue, start by checking/replacing the seat switch, make sure the block connecter is fully pushed home onto the micro switch.

look at the safety switches

look at the fuses there maybe a blown one

check your Reverse Gear engage switch and PTO (blades engaged switch), either a bad switch(es) or short in wiring indicating that you are in Reverse with the blades engaged. Is there an override function so that you can mow in reverse? Check that circuit vs. the PTO and reverse gear. good luck