For on/near the point, I place mines that fire 20+ smart swarm (size 1) missiles each. For ranged coverage of the points I have big mines that fire one or two torpedoes/MIRVs.

The best jump point defense is to sit on the enemy side of the JP. You can negate all their missile by just jumping to the other side and back.

Once they exhausted their missiles, you can let them get fairly close before sending a full salvo of missiles in their face. This should give you the overwhelming advantage to clean up with your own beam ships.

If you still do not have overwhelming advantage, just jump to the other side and ambush them.

Jump points are natural choke points of strategic value, there is no sense in spreading your assets and playing catch. And it appears that there are several ways to go about its defense:
1. Parking your feet on the Jump point and making a tactical use of jump shock effect. Using mines, fighters, missile salvos or closing in for laser hits.
2. Ignoring  jump shock and using a planet in-system for pinging the enemy and as missile lunch pad with huge reserves, and fleet assets to strike it while it goes for the kill.

The enemy will likely be using squadron transit, but the exit point is totally random, so having your ships in a central location, right on top of the jump point, gives you the highest chance of being able to zone on him.

Yes, but if the enemy force is scattered it means that he can't mount an effective anti missile defense, and you can overwhelm him one by one. Also placing yourself in the middle (of the the enemy) might not be the best thins if you are using missile ships.

I find high-speed microwave fighters work best for exploiting jump shock. It is always hilarious when a squadron of spunky fighters lock down a flotilla of unshielded dreadnoughts, although keeping fighters on station long-term does add to the logistics overhead.

Dropping the fighters on what is essentially a sustain-carrier (give it significant hangar space, crew space, ammo space if relevant, low power engines and loads of maintenance spaces) pretty much solves that well enough.
I've personally been toying with the idea of mining jump points by making medium-short range submunitions on the mines per normal, but rather putting them all in the middle, space them out so that their secondary stage ranges overlap by about half. If you're spending enough time doing so, you'll be able to make a meshed hexagon of mines that will more reliably guarantee that a shot is made on the target when it jumps in, and allow additional salvos to launch if it survives it enough to maneuver., though, they'll be vulnerable to anti missile defense, so you'd need to group them ridiculously tightly or  maybe the idea is just good for initial-strike coverage.

Quick question: do missile stages used as mines need onboard sensors to deploy, or is a nearby sensor contact within it's range sufficient? Because I'm thinking of saving resources by making it so the minefield uses one active sensor buoy and a bunch of sensorless mines.

There really isn't all that much finesse to it. The most important part is to remember to have all your ships trained to cut your own response time. The NRP always reacts quickly and I may be wrong, but their own shock effect may be lasting less than that of the player (usually less than 30 seconds) so you have to be able to fire as quickly as possible.
As for a more advanced advice, I don't think there is any to give really. The enemy will likely be using squadron transit, but the exit point is totally random, so having your ships in a central location, right on top of the jump point, gives you the highest chance of being able to zone on him and that is the tactic I myself have used. If you don't mind splitting your forces you could increase the chance of intercepting incoming enemies by having your own ships form a ring around the jump point, preferably close enough that your missiles arrive within a single five second interval (as this will make them impossible to intercept with the exception of CIWS) but this will leave your own force scattered. In the end aside from sitting on top of the jump point with highly trained and quickly responding forces, I'd say everything else is a matter of luck, as the exist point is random.