Yea, it's difficult. So I'm going to compile some tips to try and make it a little more bearable.
1) DO NOT LOOK AWAY FROM THE SCREEN. The second you do Murphys Law will kick in, something will happen and you just lost a few dwellers.
2) Map layout. This little trick has been saving me a ton. Aside from raiders and deathclaws, incidents only pass on to connecting rooms. I've laid my vault as such:
[.....][VD][VD][EV][SR][SR][EV][.....][.....][.....] Space is empty spot-EV is elevator
[PR][PR][PR][EV][.....][.....][EV][PR][PR][PR] PR is power-LR is living room
[.....][.....][.....][EV][LR][LR][EV]
[WR][WR]WR][EV][....][.....][EV]
With a layout like this, if my power room catches on fire, it won't pass on to another room due to the gaps. Also, I can hurry and evac my dwellers to a living room to save them and the fire,radroaches,etc will die out on their own.
3) Expand SLOWLY. I'd recommend not building a room till you have enough dwellers to fill it. Have a 3 tier room with 4 of 6 dwellers in it is asking for a fire to overrun and kill them.
4) Have an empty storage room or living room. Like step 2, you can always evac dwellers to in during fire,roaches, and rats.
5) People are reporting deathclaws at 35 dwellers. Be sure to have most of your dwellers leveled up and decently equiped before passing this number.
6) Sadly you shouldn't leave your explorers out if you can't check on them within a few hours (depends on level, time already out, stimpacks remaining)
7) Explote immunity. Leave dwellers outside the door, run room to room, vault door is safe from all but raiders/DC's and prego women (take care on this one as populating to fast can lead to breaking the 35 dweller threshold, spawning deathclaws and being possible under level/prepared for them.
8) Level up strength and agility first. Followed by endurance and luck seem to be a good route.
9) Normal mode you can get away with 1-2 med bays. While you still can in survival, you need to be on you A game of keeping dwellers safe. Consider having a few just so you have a constant healthy supply.
This is all I can think of at the moment.
If you have any other tips feel free to add them and share with us your progress and strategies.
**bonus question for the survival elite. Anyone know if you can repair a Mr. Handy yet?