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Justin Case Auto
Have someone hold open the throttle and clean inside the throttle body with some throttle cleaner and a tooth brush. reassemble, start vehicle and let idle for around 5 min. Drive vehicle and see if it's now ok. For future reference you should always have a battery backup installed when disconnecting the battery. On this vehicle the idle learn is really all that is lost but on newer vehicles there is a lot of reinitializing that will need to happen after a battery disconnect, trans shift points, windows, & doors, come to mind at this point but I know there is more.
Have the system pressure tested to find out where the leak is and repair it
Have someone put it on a hoist and check for worn bushings in rear suspension or something bent.
You will need a scanner that will access the anti-lock brake and traction control computers in the vehicle. If you're going to get your own scanner make sure it will do this. Most "code readers" will only access federally mandated codes for emissions. I think your best bet is to find a quality independent who can diagnosis this. It will be less expensive in the long run.
Could be a starter but have the battery tested (in the vehicle) first. Test all connections at the battery and starter.
Check for a pinched wire on top the fuel tank. The harness may have got caught when reinstalling the fuel tank.
Something is draining the battery overnight. Close all doors and make sure there is no interior lights left on. Check glove box light. Check under hood light. If all are off then take it to a shop that has the equipment and diagrams to diagnose a draw on the battery could be almost anything.
This vehicle has a fly by wire throttle system. It will need to be relearned - most likely. Unless you're experienced I would have a shop do it because it is slightly involved.
Try having the motor mounts checked
This may help. Most of the time the convertor will end up needing replacement. After replacement make sure you test to see if something caused it to go bad. Running rich, coolant leak into exhaust, burning oil. Unfortunately convertors are common on these Toyotas