Hi Eiji,
First, let me clear up at any speculation about my Hyak's neck. Yes, over time, many Hyak instruments had neck problems. However, I owned five Hyak basses and not one of them had neck problems, so my experience is very positive. Just to be sure though (since I was spending quite a bit of money), at Valentino's suggestion, I decided to go ahead with the truss rod. The instrument was already apart and the truss rod would ensure no problems in the future. If you are confident that the guitar you are looking at has a good neck, then I wouldn't worry about needing a truss rod. If the neck is warped or seems to have other problems, then you might consider adding the truss rod. Consider changing the size of your strings. On a neck with a truss rod, you simply make an adjustment. Without a truss rod, you may experience tonal problems until the neck settles-in.
The Hyak you saw restored still has the original electronics and they work great. In your case I have two suggestions-- have Alembic troubleshoot and fix them, or replace them with similar Alembic electronics.
Hyak electronics function identically to Alembic Series 1 electronics but are self-contained and built more like the electronics in the Alembic Custom Signature basses . In fact, I think the Custom Signature electronics will drop into your Hyak. The only thing you will lose is the power supply, but Ron could easily add that circuitry. Make sure you get the guitar version of the electronics.
I guess the most important suggestion I have is to work directly with the Alembic if you decide to purchase the guitar. Hyak basses are essentially Series I basses in looks and construction. Since Hyak is no longer in business, the people who know the bass best are at Alembic.
I'm sorry I can't help you troubleshoot the problems you indicated on your guitar. My electronics knowledge allowed me to make a bunch of power supplies for my Hyaks but nothing more exciting.
The Hyak electronics are active, not passive. I believe the circuitry is asymmetrical. I've forgotten a bit of terminology. Again, just think Series I (for operation).
All of the Hyak instruments I have seen (with the exception of one) required a guitar cable with a mono (normal guitar 1/4) plug on one end and a stereo (TRS) 1/4 plug on the other end. The stereo connector plugged into the instrument and a mono connector plugged into the amp. If you use a normal mono guitar cord on both ends, the neck pickup will not work. Only the bridge pickup wold work. I believe Dela217 could verify this for you.
Sorry I couldn't be of more help.
Rob