Parking
You can usually find an attended parking lot. Make sure you note when it's open/closed. You might also want to use automated parking lots (e.g. 24Times). These are often built with barriers that rise a few minutes after you park in a space. To lower the barriers you must use the machine. Input the number of your parking space and pay the hourly rate with cash or a credit card to lower the barrier.

Driving behavior
I agree with mjm's assessment of driving habits. You'll see a lot of slow, but not skilled or attentive, driving. This does not meet my definition of "defensive driving". I'm sure it's down to lazy-... instructors like the person who is teaching Taiwaned.
(BTW @Taiwaned - There are instructors in Japan who can really teach you to drive well. Fire the asshat who is telling you to just drive slow and find an instructor with real skills. Get an ex-police instructor if possible).

Driving on the "wrong" side of the road
I take it that the OP has already decided to do this so I'm not going to waste time with words of reassurance. You made a choice that works for you. Good. Here are a few of the "upside-down world" moments that you can plan for:

1) Starting off on an empty road. With no other traffic to give you cues, you might forget where you are.
Solution: Write a reminder and stick it somewhere visible. Agree a routine for starting each trip with your passenger. If you discuss how you're going to position the car when you drive off and how you're going to negotiate the first turn, you'll be OK.

2) Reversing. Woah. Freaky. ... Which way is up?
Solution: Try it out once or twice before you actually need it.

3) Turning right (across oncoming traffic). First couple of times may feel a little strange.
Solution: Find a small road to practice once or twice soon after you pick up the car.

4) Getting the correct position in the lane. Your passengers will probably tell you that you're too close to the parked cars on the left. (Or you may clip someone's wing mirror).
Solution: Think back to when you first drove. Did you have a trick like aligning the front corner of the car with the center line? Get yourself a new trick.

5) Unfamiliar controls in the car.
Solution: Ask lots of questions when you pick the car up from the rental station. Write down a checklist to ask about if you like. Where are the controls for wipers? indicators? Any trick to turning the ignition? Is there a button to pop open the fuel cap? How about the trunk? How do you tune the radio? What are the settings for lights? What does this button do? Spending a few minutes on this will save you some agro later. You don't want to have to explore the manual late one night when you're trying to turn on the headlights.

Whatever you do, don't drive when tired. That's a killer even when you're at home - much worse on unfamiliar roads with unfamiliar habits.

Places where the rules differ from what you may be used to
Of course, this depends greatly on what you're used to, but here are a few of the things that foreigners seem to get into trouble over:
1) Never turn on a red light
2) Always STOP the vehicle before crossing a railway line (level crossing)
3) Never park on the right side of the road

Insurance
Standard insurance on rental cars tends to leave you liable for some things - In addition to the standard excess amount, you can be charged a daily rate for the period the car is being repaired, for example. You might consider buying a worldwide rental car excess insurance policy. I have one because I regularly rent cars and use zipcar. You probably need to do more than one rental per year for it to be worthwhile.

Hope this helps.