Let's deal with the question of recoil first because it's simpler.
Newton's third law applies, even more so in space because there is no pesky gravity or wind resistance throwing off the effects. So, if you shoot a missile, a bullet, or anything else out of the ship, there will be recoil. This is actually critical to your spaceship's function because (when you get right down to it) the propulsion system is just a sophisticated recoil system - you're shooting out a propellant at high speed to push your ship in the preferred direction.
As for 'dropping' your missile; you can't. No gravity. What you CAN do is push it out of the ship gently, then let its own guidance and propulsion systems activate. This will massively minimize recoil effects, but won't eliminate them to a mathematical zero.
Now; let's get into point defence.
Yes, for the reasons described above, the missile WILL push you in a direction on impact, but that's not the primary reason for wanting to avoid it hitting you in the first place. whatever the payload is, it's dangerous to your ship otherwise the enemy wouldn't be investing the energy and cost to launch it at you in the first place. HOW it's dangerous isn't material to your question; let's just say that none of us want that missile to hit and we want ways to stop it.
Modern warships and some modern military transport planes (including Air Force One) have a range of what we call point defence systems for getting rid of missiles. They range from starburst style hot chaff systems designed to get the missile to detonate before it reaches the plane, to 'metal storm' solutions that fire a massive volley of (essentially) ball bearings into the path of an incoming missile to (again) get it to detonate before reaching a ship, to turrets and guns designed to shoot down the incoming ordnance. One can also consider drones (both UAV and subs) to get in the path of torpedoes and other heavy ordnance so that the primary target is protected.
Your spaceship is no different, except that it has to consider a 3-dimensional theatre instead of a 2-dimensional theatre. Your best bet is to have turrets strategically positioned around your ship, and a small supply of kamikaze style drones that are programmed to fly directly into the path of incoming missiles as a contact target, protecting the primary ship.
To address Joe Bloggs' comment...
It has been pointed out that the point defence system may only be part of your solution because the overall inertia directed at your ship may not change much, just be spread over a wider area based on missile debris (Thanks Joe). Two your point defence system should, therefore, be designed to prioritize payload destruction, meaning that the 'dangerous' part of the missile (the warhead) is inactive by the time the missile strikes, and you're dealing with just the raw impact, which your armour would ideally keep out.
Of course, that means missiles would evolve to keep the payload as protected as possible, meaning that the other strategy would be for your drones to be of sufficient mass to knock the missile off course. If that was their sole function (especially if the payload is designed NOT to detonate until near the primary target) then you'd employ the 'double tap' of knocking the missile off course with your drone, then taking it out with point defence turrets.
Ultimately, you want the primary ship protected, so your only choices are to destroy the missiles before they reach you, or provide another target. Turrets and drones give you options that cover both strategies and with the addition of the above strategy, also deal with the 'mass driver' style missile system.