Against Mage/Rogue, you have your trinket to safely live one go. You have healer’s trinket to then safely live another go (order may be the opposite depending on comp).
After that, it gets to the point where you will need to preemptively use a defensive (Netherwalk, blur, darkness, …) or a setup disruption (such as imprison, or eruption) before one of their go in order to live. The “flashy” plays.
It seems hard and “RNG” but it’s actually not that “RNG” if you know how to recognize a setup. Most people don’t play with voice, so they have rules on how to setup with eachother that they try to hold on to, so it’s actually kind of predictable with some experience.
A clear sign that a setup is coming on you: stun DR on you is available (0/3), and poly DR is available on healer (0/3). As soon as DRs are cleared, it’s a good time to expect a setup.
A good clean setup on DH+Healer from Mage/Rogue usually looks like this:
Dragon’s Breath on you at the same time as the rogue stuns your healer, and then the mage polymorphs off the stun on your healer and then the rogue comes to stun you after DB. The DB is for you to not be able to preemptively use something like Darkness, Nether, Blur or one CC for the mage before they stun you.
It’s telegraphed in the sense that you’ll see the mage stop anything they are doing and go in your face, or at blink range of your face and they will not cast a Pyroblast or anything so they don’t have dots on you. They could also polymorph you into repolymorph on the dispel for the exact same purpose, if they didn’t have to use poly DR on you for running away/keeping alive between goes (this usually indicates a lack of pressure from your team between goes).
I’ll only describe the tip of the iceberg, because we could go extremely in detail about this matchup. And as I don’t play DH, you should look for the advice of someone who mains it for sure, I only play Mage/Rogue :D.
You have a few choices here when you recognize a setup. Most of this will come from experience anyway, don’t worry too much about it.
You could at this moment when you recognize a setup, just imprison the mage and delay the go, or maybe they are not that well coordinated and the rogue will stun too soon (you will see it on BigDebuffs or something) and you can immediately imprison the mage or use a defensive (or your healer can too). An important thing is of course staying in combat for both you and your healer so there’s no random sap coming.
A safe call is when you recognize a go, just use a defensive. You absolutely need to track enemy combustion too, and shadow blades. They are the scary goes. These IMO warrant trinket or the very strong defensives. At least for combustion.
Also remember that interrupting polymorph cast as much as you can is crucial, even if you’re hitting the rogue (using focus macros and even sacrificing mobility for it).
So, to summarize, what you should take away from this is the “beginner tier” of dealing with Mage/Rogue: keeping very close attention to what happens to your healer and as soon as you see a stun or anything, just react to it, like imprisoning any of them. And keep attention to the mage who suddenly instead of running wants to go to your face (he wants to DB you for setup).
Next tier is actively watching your own DRs to know when you’re vulnerable or not. Same for DRs for your healer.
A last thing for the “beginner tier” is to also not focus too hard on “surviving” or being defensive vs M/R. If you put no pressure, they will be able to do as many goes as they wish and eventually you will have nothing left to not die and will die even to shadow dance and ring of fire. A good moment to pump offensive CDs for pwnage, is when sheep DR on you is 3/3 and stun DR on you is 3/3 - nothing they can do about you unless they have a major CD like Vanish or IceBlock. Generally once you get rogue’s trinket and a good stun on him it’s enough to make unrecoverable damage.
Of course you want to take as many defensive talents as you can for the specific Mage/Rogue matchup. So, Rain from above (a pre-rain will basically remove one of their goes), netherwalk, glimpse, and of course, a CRUCIAL talent vs Mage/Rogue: Reverse Magic. This will literally win you the game vs setups that are too rushed ending up with gaps between stuns on you. A small gap on a stun on you can end up with a fel rush + reverse magic on the polymorph and giving you a 2vs1 on the rogue.
Of course all of this is so situational that the best you should do is just keep playing, recording your games and just keep posting videos if you need some advice on what you could have done.
And lastly, your healer has a big part on you winning this matchup, it’s a team effort. There’s only so much you can do alone to carry it, but it’s good to always try your best to utilize all the options available to you, at least as practice. All that I have said about preemptive usage of defensive or abilities (like Greater Fade, Blessing of Sacrifice, SWD etc) applies to healer too.