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Wow. $500 is a decent price for an 8th gen, even with high mileage. Though it does make you wonder if the previous owner was trying to just sell the car quickly.
The P0446 - I'd reset the CEL and see if it comes back. Many of the EVAP related codes on the 8th gen can be spurious errors, not even related to the EVAP system.
Engine noise could be from a number of possible culprits. Since the noise seemed to disappear after an oil change, I'd look into getting the used oil tested. Check for excessive wear metals, coolant in oil, etc. Choice of oil filter is also important - if the car hasn't properly maintained, possible that there are heavy engine deposits/sludge inside that is clogging the oil filter quickly. Once that happens, the oil filter will go into bypass mode (ie, not filter anything), and can cause funny, rattling noises. In the mean time, keep an eye on the oil levels - checking daily before you take off for work is not excessive. At the very minimum, check weekly.
Shaking and bouncing at speed could be from a number of things. Could be a bad tire, bent wheel, bad wheel balance, poor alignment, worn hubs/bearings, loose brake components, worn suspension, worn axles, or damaged transaxle. Try rotating the tires front to rear - see if that changes the shaking. If it does, it is likely something with the tire/wheel set (balance, bad tire, bent wheel). If the shaking stays the same, then it is fixed to the chassis (brakes, suspension, hub, axle, transaxle).
OEM tire is 175/65-14, optional tire size is 185/65-14. If you rubbing, could be either wrong size tire, wrong wheel offset, or worn/damaged suspension. I'd also double check the chassis - see if a shop will put it on a jig and see if the unitbody is straight. That would rule out any accident causing some of these issues.
Same with the springs and struts - if you hit a bump and hear the car rubbing or bottoming out - could be heavily worn suspension or damaged suspension components. Measuring the static ride height is a quick and easy check. If the car is tipped down at one corner, that could indicate a broken spring. Check each corner carefully, check for tell-tale leaks on the strut assemblies - that is a sure sign of a blown strut.
As for the grinding in gear or gear popping out - I'd drain that transaxle ASAP and see if you can see any chunks/debris coming from it. Make sure to crack open the fill port first, before you drain it.
Sounds like this car is in desperate need of some care and maintenance. If this engine happens to not consume oil - chances are with just normal maintenance, you can get quite a bit of life out of it still - as everything else you mentioned is just a maintenance/care issue.
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2002 Corolla S, 1.8L 1ZZ-FE VVT-i
2003 Matrix XRS, 1.8L 2ZZ-GE, VVTL-i (RIP)
2009 Matrix XRS, 2.4L 2AZ-FE VVT-i
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