Just an FYI, when they ask to have access to your computer, what they're asking for is 'Remote Access'. It's a legit service offered by many technology companies, and it probably could've helped them find a better answer for you. You had every right to refuse it, but I'm just saying it probably could've helped them more. On that note, I really don't see, in all honesty, how remote accessing someone's computer will help them identify an issue with a printer being that a printer is an external device and is in no way related to the computer except for when you print from it. I'm going to take an educated guess that what he was trying to look for was your Page Setup options and ensure that all of your settings were correct.
I think having to e-mail "proof" is going a little too far and shouldn't be the customer's responsibility when the fault lies with the vendor. I would think, at most, the customer should only have to provide proof of purchase to validate that the device is within warranty limits, and offer a replacement printhead. Are you certain that cartridge is an inkjet cartridge and not a printhead cartridge? Sometimes the two can be confused to look like one another.
As far as the $4.99 cartridge is concerned, I wouldn't buy it. A cartridge that is THAT cheap means that the printer is a POS (And I'm not referring to a Point of Sale). You're looking at inferior power, speed, and functionality as demonstrated by your experience; plus, I'm not that big of a fan of Lexmark. I own a different brand of printer that works pretty good.