unlikely for a new tranny
reverse gears are spur cut gears so that means that if the teeth are not properly aligned they hit on the ends of each other stopping engagement
you can get around it but selecting neutral , letting out the clutch the depressing the clutch again and make the selection
that sets the gears spinning and makes for alignment for selection
The circuit you hear powering up for two - five seconds should be the fuel pump priming the fuel system . What type transmission ? manual or automatic ? Do you know what a wiring diagram is an how to read it ? Free wiring diagram's here http://www.bbbind.com/free_tsb.html Enter vehicle info. Year , make , model and engine size. Under system click on engine then under subsystem click on starting .Click the search button then the blue link . If you look at this diagram you will see the fuse at the top of the page that powers the ignition switch . You will see the ignition switch , the starter relay and the PCM - engine computer ! The PCM supply's a ground to energize the starter relay . But only if it see's that the transmission is in park or neutral ( automatic trans ) Or the clutch switch is closed ( manual trans ) .To see this you would need to go to transmission controls under transmission/transaxel . Plus the PCM will not energize the relay if there is a anti-theft system problem . You could do testing right at the relay base (where the relay plug's into ) With a test light ,voltmeter etc... Watch this video on relays
If you mean you want to start the car without the switch interfering, yes, you can do that by cutting the tan wire from the starter relay and grounding it to the chassis. This will allow the engine to start in any gear. However, the switch does a lot more that neutral safety. It tells the pcm what gear you want to choose. So, if the switch is bad, it could give the pcm bad data about your selection. Bottom line is, the car should start (please have your foot on the brake when you try), but the transmission may not work as expected.
The mechanism is called a window regulator and you have to replace the entire unit if a plastic piece broke that supports the cable. The mechanism is bolted to the door. alldatadiy.com is a great pro grade shop manual.
There is voltage on all three wires because there is no ground, check with engine cranking. The two wires that go to computer are grounds for coils. You could probably use a noid lite to check for pulse, engine cranking. That should be safe, between primary voltage and one of the grounds. There several ways to visually check for a pulse. If your digital multimeter has a min/max setting, that should work. You could use an l.e.d. but you would have to use resistor.
You changed computer, was it new?