Car shake when running hot when ideling. Worst when give slow gas. Smoother as gas pedal press faster.

There are a few possibilities;
First the timing belt is critical to replace on your car. You have a, "Interference" type engine, which means if the timing belt fails, the valves will hit the piston and cause catastrophic damage.

As far as the idle, I would look at the O2 Sensor, the MAF sensor could be in need of replacement, Also Exhaust Gas return (EGR)Valve, Idle Air Control valve, (IACV), as well as other emissions or intake sensor items dependent on the vehicle. I cannot recall if your car has a Intake Air Temp sensor, however, the overall issue sounds like the entire intake is probably in need of cleaning which could easily be done when the timing belt is serviced. The intake gets so dirty after time that it is hard to pin point which sensor(s) are the culprit.
For right now, I would replace the timing belt, and take care of other service items that, "Overlap" in terms of labor. While replacing the belt, the water pump is easy to replace, the belt tensioner, cleaning the intake completely, it all depends on your budget.
You have a rare and cool car. The timing belt is way overdue and replacing it and cleaning up the intake, etc. Is probably the best place to start. I can try to put an estimate together for you for a timing belt. The overlap in labor kind of depends on your repair shop and other factors. I would be glad to help you with that as well.
I will keep an eye on this post and get back to you ASAP when you replay.

Thanks very much. It is always great to hear when the work really helped someone. That, "Thank You", is the best donation you can give.

The following estimate is to replace the timing belt and water pump. I do not know how much overlap labor there is for other items you might ask the repair shop you go to if they can help with that to have a good, "Plan of Attack" for your car. If you don't have a repair shop you go to yet. This site can help you find reputable repair shops near you.
Some pointers for finding a good repair shop;
Does it look clean and reasonably tidy?
Do the employees carry a professional attitude/ e.G. Do the techs wear t-shirts or uniforms?
Is the equipment in the shop old and outdated or well kept and not outdated?
Try to find a shop that pays its' Techs a salary wage instead of just Flat Rate.
Lastly your first instinct is a great sense for anything that you have to have some faith in.
Once you find a shop, stay with them. Car repair is tough, things sometimes happen and even the best shop and or tech, can make mistakes. The advantage to being a "Loyal Customer", is the give and take always comes back around. Repair shops do favor customers that try to understand how tough car repair is.
There is an old saying about the cost of car repairs and how people complain;
The human body has not changed in thousands of years. Cars change constantly and they are more variations and manufacturers every day.
So, why don't people complain about their Drs' bill like they do their repair bill?

Here is Estimate;
Approximately 6-7 Hours of labor @ $75/Hr, with parts costing $200.00=$725.

Thanks again and let me know if you need anything else.

I am in Alabama near the Florida border.
I just moved from Maryland near DC.
Good luck. You have a really good car to work with.
Eventually, if you want to modify it, try to stay near stock, but make it the best it can be. Stuff like port matching and some tasteful mods that are not always cheap, will lead you to a fun and dependable ride.
Stillen is the best with Nissans. The money is worth it. Also join an enthusiast forum.
Having that car running pretty much stock with small things like matching ports, etc. Will be a lot of fun. You can get the twin turbo kit from Stillen. That is not too far from stock. Even then you might have to change more than you think and I had a friend with a TT 300ZX and it was a handful to drive. The motor you have is bulletproof as it was made to be boosted.
Make a plan for what you want and research what you will need to get you there and how reliable the car will be if you stray from stock. You have a really cool ride and I am a little envious.
I used to have a 2002 3.5 Altima. It was too fast and too heavy.A few accidents here and there lead to my insurance going up and me selling it. I made the worst mistake in getting lowering springs for it. They actually made it handle WORSE and it caught the exhaust on a man-hole cover and ruined $2000 in parts.
I was happy to put it back to stock. So, just keep in mind that straying from stock has to be well thought out.
I look forward to seeing where you take this one to.
Take Care.

Dr. C

I have the need for more power as well. There is nothing like fixing your own car. It is very satisfying as well as stressful.
Twin Turbos would be some work, but since it came as an aoption, it makes it a lot easier to do as opposed to a car that never had that option. Join a forum for 300 ZX owners. There is a bunch of them and there is wo much information from other owners who have experience with modifications, repair, how tos, etc.
Port matching, for an example, is when you take two intake components and match the flow paths by grinding them and making them both the same shape. So, the intake manifold and the throttle body would not have any difference in shape inside were the air flows in through the mating surface as if the two parts were one smooth continuous port. This includes cutting gaskets so they don't stick out and impede the flow of air.
Don't mess with the ECU just yet. Join a forum, see what is out there and what works the best from other owners who have already done it. Look ay the Stillen web site and see if there is an enthusiast link there. Stillen is a very reputable aftermarket company. Steve Stillen used to be the factory race driver for Nissan and still helps in development.
I am 41 and as irresponsible as ever.
Get your car solid as far as getting the maintenance and stock platform, solid and then start thinking about where you want to be if you did everything you want to it. Don't skimp on good parts as cheap stuff always end up costing more in the long run. You would be surprised how well your car will run with some port matching, an aftermarket intake, (Nissan has a line of performance parts called, "NISMO". Not cheap but excellent quality), a header and exhaust and getting basic maintenance items such as the manufacturer suggests. Staying near the stock design makes things easier, but you can modify as long as you do the research to find out what works.
A good an example is a drag race Prius that Bisimoto engineering got to run 9 second quarter mile without using a supercharger, a turbo or nitrous.
Thanks for the acknowledgement for the advice. It is always a good feeling to know that working on this site helps people.

Take care and keep me posted.

Go Buckeyes!
Talk to you when I hear from ya :)