Welcome to the forums. Right now your Corolla will either have a 1.6L 4A-FE or a 1.8L 7A-FE. The best way to check which engine you have is to look at the engine code that's etched into the block. It's located to the right of the exhaust manifold and is stamped vertically.
Here's some photos of it's location:
As you can see, mine says 5A as it has a 5A-FE engine. It will says 4A for the 4A-FE or 7A for the 7A-FE. You will have to get down low to see the numbers on the block and may have to give it a bit of a wipe to get rid of any dirt etc, but this is the most reliable way of checking which engine your car has.
As for the wiring, the 1.8L 7A-FE has a knock sensor whereas the 1.6L 4A-FE does not, so the loom is slightly different between the two. The ECU will also be different. The engine mounts are the same however and the transmissions are interchangeable. In regards to getting a JDM engine, bear in mind that the JDM engines didn't have an EGR system as the emissions standards were different between the US and Japan, so if your engine currently has an EGR system then that's something to bear in mind.
I'd suggest replacing the motor with one that is identical to the one you're taking out, just to avoid any problems and additional cost. Most wreckers or junkyards should be able to sell you a replacement engine as both the 1.6L and 1.8L are very common. Good idea to do the timing belt replacement while the engine is out. It would be best to also replace the rubber hose that leads from the water pump to the block. Replacing the valve cover gaskets is a good idea too, but this is an easy job and can be done once the engine is fitted if need be.