NRT to HND?
 

I have the following itinerary:
LAX-NRT arriving at 15:25
NRT-SYD departing at 22:20

Is it true that I will have to personally take my checked luggage from Narita to Haneda? Any recommendations for this transit? Finally, any quick, recommended sightseeing to do along the way?

Presumably you mean HND-SYD (not NRT-SYD, otherwise this is a non-question)?

Luggage - yes, you'll almost certainly have to take your checked luggage from NRT to HND. It may depend on who you're flying and whether it's all on one ticket but it's very unlikely that you can avoid doing that. You'll be "entering" Japan so your luggage goes with you....

To get to HND from NRT, take the limousine bus. As for sightseeing along the way, there's "Tokyo" but apart from that, no....

I've never heard of an airline moving your luggage between the airports. They are not close at all.

You have time, so you can take the limousine bus, as that is easiest. Cheaper is taking the Keisei train. I think there is a direct one.

You will need to carry your own luggage between the airports. Limousine bus is the most convenient for you, but it willt get you to HND a couple of hours earlier than you need to be there.

You don't have a heck of a lot of time for sightseeing in Tokyo, but if you wanted to do it, you could take a train to Ueno, Tokyo, or Shinagawa, then leave your things in a locker for an hour or two before continuing onward to HND. Out of the available options, Shinagawa is probably the least interesting to look around, but the simplest, since you can take the Narita Express from NRT to Shinagawa and a Keikyu airport train from Shinagawa (Keikyu station) to HND.

If it were me, I'd just take the Limo Bus straight to HND, check your bags in first and be done with it. Chances are, you'll get to HND by 6pm. You may get there later, or maybe much earlier if your plane arrives early and there's no queue at immigration. Make your plans depending on what time it is after you've checked in your luggage.

If at that point you have >4hrs till your connecting flight, then I might head over to Kawasaki. Make sure you take the right Keikyu train. It's only ~15min ride. Kawasaki has tons of food, tons of shopping. 2hrs there will go real fast. Actually there's so much around that area that, if you're not familiar with the area, you might spend quite a bit of time just trying to orient yourself. The DICE Building right by Keikyu Kawasaki Station is a good spot with Tokyu Hands, bookstore, Uniqlo, drugstore and restaurant arcade. The underground mall connecting Keikyu Kawasaki with Kawasaki Station is even better, and that alone may take up 2 hours. But the restaurants there are hit and miss, and you need to go to the right one. The crown jewel of Kawasaki, though, is the Lazona complex which is a good 10min walk through underground mall and the JR Kawasaki Station provided you don't get lost.

If you have <3hrs, I'd stay put at HND and just enjoy the landside mall & restaurants and the observation deck. The airside facility at HND Int'l is pretty enjoyable actually, and there's no price gouging at shops either. I presume you're flying NH. If you're flying in premium cabin, then be sure to save some time to enjoy the lounge. Even if you had >4hrs, I'd suggest you just hang out at the airport if you're tired. I think you will be, due to the time difference.

Second that! It is easier and faster. However not sure if one will be able to check in and most luggage storage may be full in early evening already (and will have to be redirected to delivery counter for temporary storage)

I will recommend Sinagawa too. 15 minute ride every 10 minute.

The limousine bus is the most direct way to get from point A to point B. But you've got seven hours between flights. Assuming it takes you a half-hour to clear formalities and you want to be at Haneda an hour or whatever ahead, that gives you some time to muck around. I'm inclined to tell you to take your stuff on the Narita Express into Tokyo (or Shinagawa), put your stuff in a coin locker there (700 JPY for a big one), and then go take a walk.

Tokyo station is near the Imperial Palace, which is generally closed to the public with the exception of the East Garden (which is closed Mondays and Fridays). You're also near the Tokyo National Museum of Modern Art (also closed Mondays, 420 JPY). I can't offer you any personal recommendations for eats there. Research something, or maybe stumble upon something.

To Haneda: Take the Yamanote or Keihin-Tohoku line to Hamamatsucho and transfer to the Tokyo Monorail OR take aforementioned trains to Shinagawa and hop the Keikyu Airport Line OR take a limousine bus from Tokyo Station (departs every 30-60 minutes). The train options are the faster ones.

Well it depends on what you end up doing.
You can get the Limo Bus ticket with credit card (as long as you're buying the ticket at NRT, HND or TCAT). If you end up deciding to head out to town after you check in and drop your bags off at HND, then I'd recommend withdrawing at least 5000yen in cash. For the most part, train ticket machines only take cash or IC card (Suica, Pasmo... which in turn most likely requires cash to obtain and to deposit value onto). If you wanted beverage from a vending machine, again only cash or IC card. Most stores and eateries accept credit cards, but some don't.