Your question is not "hard to answer" as you state. You must realize that if you are looking for FREE information/advice from working professionals in this field you must be patient. These forums are are great resourse and very helpful but if you wish immediate responses to questions try calling an NEC authorized dealer.
I only look at this forum once every couple of days to see if I can help anyone out as I'm sure many others do so don't assume that your question cannot be answered because you did not get a response within 24 hours.
As for the Gate Keeper the NEC supported/tested product is the NEC GK1000 though I have heard of people getting successfull connections via Cisco and Nortel gate keepers. There many Gate Keepers out there but H323 never did become the big thing it was originally made out to be and NEC is placing their eggs (like every one else) in SIP so the H323 technology is very by the way side and rarely do we see much of it anymore.
H323 never took off due to the lack of features available using this connection medium. No name display, call forward with VM integration, no call back, no voice announce, and the list goes on and on.
The only advice I can give you on this one is try it but that would be a pretty expensive test if it didn't work. I'm guessing the cards and software to make up to 4 simultaneous connections (minimum of 4 H323 trunks must be installed) will be probably the best part of 2K maybe even more. I don't know much about the pricing side. I just design, get it working and then support it.
For the same price (again I'm guessing) you could probably get an NEC SP20 or SP30 operating on your PC and retain all the features of an origianl digital set. Spending money on this route is going to get you results as it is a supported app that is installed in 100's maybe 1000's of sites around the US and even more around the world.