I will answer what I can without repeating the good advice that you have already received. My current AAR is designed specifically to help new JFBs get started playing Japan.
2: Industry / Aero Engine Research
I have no idea how to efficiently rearrange the number of engines I am producing and "researching" at the start of the game. I think the more advanced (future) engines I produce, the faster those engines become available, or something. Or maybe that's only for airframes? And if that's true, I also have no idea how many engines to produce to research efficiently. Is there a formula or curve for this? I don't want to allocate too many or too few HI. Also what is the main downside of simply expanding a lot of HI at the start of the game?
You want to base your engine research and production on your airframe research and production. So plan airframes first; then engines. Do as much as you can at game start. How exactly you structure your production and R&D programs depends completely on whether you are playing PDU:on or PDU:off. I recently posted most of my fighter R&D for my PDU:off game with Apbarog on my AAR.
3: Manpower / Pilots / Training
Also, pilot training missions seem complex. Is there a general rule for what skills are most critical to train, and what training levels are good/optimal? I gather that you want to have an air HQ and lots of obsolete aircraft flying training at a certain altitude with a certain % of the pilots resting. Any suggestions for extra units to send to Manchuria for training purposes?
What Getassista said: air and defense are keys for Japan. Defense trains best on low altitude missions: strafe, LowN, and LowG. I recently posted excerpts from my fighter pilot training program on my AAR from Apbarog in which I employ most of the best practices for efficient pilot training.
6: "Fortress Palembang"
Do Allied players still do this? This seemed like a killer strategy at least back in the day and I'm not sure how to beat an Allied player who simply stacks this hex.
Palembang must be one of Japan's first objectives: even when playing DBB where refineries do not manufacture supply. Not only do you want to avoid a buildup of allied troops here, you also want to avoid a buildup of allied engineers, which can damage the oil and refineries during any battle. How you deal with Palembang will vary based upon your overall strategy for the opening move. In my two recent games, I moved KB to the Singapore area on Dec. 7th. This effectively ended the fortress Palembang threat as every allied ship fled from the area or was sunk within the first 3 days of the war. If you do not move KB to Singapore, taking Singkawang early, as early as the 2nd turn, is very helpful: especially if you can bring in an air HQ immediately. This will allow you to threaten any transports that move there and bomb the airfield daily until it is closed, so that engineers and troops cannot be airlifted in from Java.
I don't see many AAR where the JP player seriously attacks India. Is there a good example of this? They seem to love going for the Aleutians and North/West Australia. I have no idea why the Japanese player would target these locations. Calcutta seems like a juicy target with 500 HI.
What you do, where you focus your efforts, and how far you go in any theater should vary from game to game based upon your initial plan and opportunity. If you are able, for example, to destroy a large number of British and Indian units and achieve a significant advantage in this theater early, then why not reinforce success and push into India to try to capture some supply? The same applies to Australia. You are limited by fuel and supply, however, and do not want to over extend yourself. The most important thing as the Japanese, in my opinion, is not how far you advance, but how many allied assets you destroy. If you can destroy a large number of allied assets without going into India or Australia, you may have no reason to extend your perimeter. If you feel that you must enter these areas to provoke a fight before the allies are too powerful, then do so.
What is the general strategy for conquering China?
Most JFBs believe that conquering the whole of China is essential. I do not see it as such. I think that you want to create as small a perimeter in China as possible and seize what supply and resource generation you can, but I do not think that pouring unlimited supply into China is advisable. Others have a different opinion, so take mine with a grain of salt.
9: Pearl Harbor Second Strike / Pearl Harbor Exodus
When testing, I placed the Lexington inside PH on turn 1. As the Allied player, I placed it in a task force and told it to leave the harbor. When the KB strike arrived at PH, the Lexington was at sea. This was with the "Dec 7 surprise" setting turned ON. Is this working as intended? It seems that the allied player can simply move his ships out of PH on turn 1 to avoid the strike (or even attack the KB).
Also, is it generally considered advantageous for Japan to strike again on 8 Dec or after? It seems to me like there is potential to finish off a lot of heavily damaged BB.
Finally, I notice a lot of my J aircraft are dying to flak on 7 Dec, especially the ones that attack airfields. What altitude do you suggest for airfield attack?
Striking Pearl Harbor grows less and less fruitful with each new iteration of the game. There are advantages to the Pearl Strike, but disadvantages as well. Follow up strikes will very costly in terms of planes and most importantly, pilots. Generally, these strikes produce less dramatic results than you would expect. PBEMs generally include a house rule against forming new TFs on turn 1. The AI will leave his ships in Pearl for you.
In my opinion, the major reason to perform follow-up strikes on Pearl is not so much to destroy BBs, rather it is to destroy U.S. air assets, with BBs as a fine secondary prize. The cost in terms of your own pilots and planes is high, however, even if you mix you fighter sweeps with bombing attacks for several days after the 7th.
10: Good Sub locations
Where are the best areas for J subs to go in order to sink convoys? I really have no clue.
Against the AI, you subs will cause havoc in and around Pearl and in SoPac around Pago Pago and Suva and Christmas Island. Against a human player, it is much more difficult to have success with Japanese subs. If you go into the Aleutians, you can use your fleet subs to some effect along the West Coast. Otherwise, you are probably best sticking to the areas around Pearl and SoPac, in the Indian Ocean along the Coast, and near Perth.
13: Facilities Expand at Start
What is the cost of expanding? Is there any? Do people use this option or do they manually expand only the things they need?
Mannually expand your facilities and do so judiciously. Expanding facilities is supply intensive and this is supply that you cannot use to fly air sorties, replace casualties, upgrade devices, and conduct operations.