"Where is a good anchorage just outside of the East River to prepare for the trek through the Gate? What is the optimum tide time to go through? "
Best place is simply "Little Bay" adjacent to the Fort Totten USCG Station, just east of the south ramp
of the Throgg's Neck Bridge. No facilities--but it is a general mooring area designated as such, for vessels holding for the tide.
You'll want Eldridge or WXTides32 (free software
to download) to check the slack CURRENT
in the Hell Gate on any particular day.
There's another holding area just northeast of Roosevelt Island on the other side of the HG, (Potters Cove??) but the bottom is foul and that's no place to stop unless you are transiting northbound.
Once you go past the HG and south on the East River, there's no reasonable anchorage until you are outside NY Harbor. You might go into Gravesend Bay (doesn't appeal to me) just SE of the Verrazano but if time allows, you've got much better shelter inside of Sandy Hook, just watch out for commercial and military traffic coming out from the Jersey shores.
Reasonable moorage? Well, not south of City Island, although many clubs there offer reciprocity.
Dockage? On the NJ side of Manhattan's Battery
area, but "reasonable" is a subjective concept
Unless you need to tie up, I'd go for Sandy Hook and then pop out from there. Or Gravesend Bay, if the wx was good but you just needed a short stop before continuing on.