I can't (usually) ignore a true optimist……or are you simply way out of your depth? This is not really a "faulty" lathe by the sound of it so much as a seriously smashed up one!……
Firstly, I suggest trying to see if you actually have a viable lathe that can be put back into useable condition…….all the 'faults' that you have described might just indicate an 'ex-lathe' that is no more in real practical terms….So, first I sugges getting the lathe completely apart so that you can examine and 'prove' each component…if you can't see the basic structure of a lathe potentially working well thn you, frankly, have nothing that will be of use to you in the future,…you might simply have to sell off the bits and start looking for another lathe.
Get the thing down to it's basics,…aquire centres for both spindle throat and tailstock and check that they line up exactly,…..examine the spindle bearings and their function to make sure that the headstock is not bashed out of alingment,……..remove the leadscrew and any toothed wheels while you check these basic functions,….see if you can straighten out the leadscrew by CAREFULLY reworking it using a hammer with leadscrew between two hardwood blocks,…look down the length of the leadscrew and keep tapping away until you can't see any deviation along the length……..do same with the saddle screw and slide screws,….make sure everything is nice and free and well lubed, then start to look at the electronics,…first test the motor for function without the electronics if poss.
You might want to remove the chuck cover in it's entirety as it is only there for effect and the lathe wil be much better in use without it anyway! Works steadily thu the lathe examing and proving each part…ask here for advice on each part you are unsure about….Good Luck! you are going to need it!