If you haven’t guessed already, the reason why there are several ways to chain your effects together to process an instrument or a vocal is because there is more than one way to skin a cat. There are those that advocate “there are no rules” but I don’t feel that’s entirely accurate… or helpful. 
Like many other things there are pros and cons to where you position a Compressor within a signal chain. The way it and the eq interact are one thing to consider, as is the way it interacts with a gate.
When I was first mentored, way back when, the engineer had one very simple principle concerning eq. I’ve since adopted it to this day. “Boost what you like, cut what you don’t like”. It makes total sense, right?!? 
Extending on that, it makes sense to cut what you don’t like before you compress. That way you’re not exaggerating the issues. This is what I see you doing with your Hi-Pass and Deesser. Whereas the Gate is rejecting signal based on dynamics. That all seems like a very sensible setup. In film they generally do the same approach with their ocular grading, which is effectively the visual equivalent to mixing for us. They “correct” before they “enhance” and get all creative. 
Where to put the gate in relevance to the compressor probably poses more of a dilemma. Since they are threshold based, they can certainly benefit from a more consistent signal so you can set it more reliably. This would suggest maybe putting the compressor before it, like you were thinking. However, the trouble with that since the signal is being compressed it there are less dynamics making it harder for the gate to determine what to keep and what to throw out. If you set the compressor up after the gate its less of an issue, but your signal varies more radically making it difficult to find a “set and forget” position. Either way is a valid approach, they just don’t come without their problems. 
Subtractive eq before the gate could help to eliminate false triggers, but probably only a tiny bit. A gate with a sidechain eq could probably be more helpful so you can select a consistently dynamic part of the vocal range to have it trigger. In fact a compressor with a sidechain eq can also help. Providing your compressor has a sidechain, that is. For other instances where the gate may be falsely triggering to not triggering, you may have to put a little automation on your track, raising or lowering the threshold automatically.
If you find a compressor or limiter over or under doing things in a particular part, shifting the threshold accordingly may help with that.
Unfortunately, though, there is no “correct answer” in relation to compressor positioning. What I suggest is trying multiple methods, at least at the beginning. Take the same vocal track, process it one way, copy and mute it, try another way and then maybe try even doing a third configuration. Once you’ve ears have had a break, maybe overnight, come back to your attempts, level match them and compare. What works, what doesn’t?? Maybe try too improve on them, then compare again. Do this a few times for a few different songs and see if you can come to a system that works for you.
If you do end up doing this, let us know your results as I’m sure a few of us will be curious to find out.
If you have access to Scheps Omni or StudioRack have a close look to see in what order the “pros” decided to do things and how you think they managed to get past the issues with the noise gate threshold, if they even use one.
