Modern cars absolutely hate having low battery voltage. It throws everything into chaos and can easily produce the sort of problems you’re having as the major components of a car are, these days, all controlled by computers.

By turning the car off and on again, you’re effectively performing the old Alt-Ctrl-Del function to reboot a home computer or laptop. As such, and especially since the transmission works perfectly when you reset it, this is almost certainly an electrical problem. A more permanent reset might be needed, so try leaving the battery disconnected overnight and then reconnect it next morning. This will often return things to normal.

A squeak when taking off from rest could be anything from a brake problem to a wheel bearing fault, a noisy engine drive-belt, bent axle, a dry suspension bush, worn universal joint or just about a hundred other things, including the driveshaft you already suspect.

Of potentially greater concern is the cut-off exhaust system or, more correctly, the previous owner. Anybody who cuts up an exhaust to make more noise is also a likely candidate to have driven the vehicle hard. It might be better to find a vehicle that hasn’t been messed with and potentially abused.

A sawn-off exhaust system shouldn’t harm the turbocharger or engine, but it can make a difference to performance due to reduced back-pressure. It also makes me wonder what else has been changed of fiddled with in the name of more power. Sometimes the performance will actually be improved, but you also open a can of legal and insurance worms by changing things like factory exhaust systems in what is a fairly agricultural way.

It's quite probable that the check-engine light was triggered by the worn out condition of the battery you replaced, not the brand new one being fitted. Such cars hate low voltages, and that's precisely what a worn battery produces. That said, changing the battery can sometimes cause a modern, computerised car to throw a fault code, and when that happens, the check-engine light will illuminate.

You can try to re-set the vehicle's computer by disconnecting the battery and leaving it for an hour. Then reconnect the battery and see if the light has disappeared. If it hasn't, it may be that the computer simply needs to relearn a few parameters after having been disconnected from the power when the original battery change was made. A decent drive might return things to normal.

But don't forget the basics: Check the condition and cleanliness of the battery leads, terminals and make sure that all the connections are tight. If there's still no joy, have the vehicle electronically scanned and let the computer tell you precisely what's wrong.