Coming back to the original question of backward compatibility, I think there are few different questions here. Here's how I see those questions and the answers:
Question 1: Suppose I have an old-school (AD&D, OSRIC, BD&D, etc.) adventure. What happens if I run it straight out of the box, using the statblocks strictly as written?
Answer: You can't. Your monsters have no ability scores except for Int, so you'll run into problems the first time one of your monsters has to make a non-Intelligence saving throw. And your ACs and attack modifiers go the wrong way. And your monsters' special abilities don't have save DCs. You'll have to do some conversion to get over these mechanical hurdles.
Question 2: Say I come up with a simple conversion template: Armor class = (20 minus AD&D AC), attack bonus = (20 minus THAC0), Dex save bonus = (20 minus Breath Weapon save), monster save DC = (10 + monster Hit Dice), etc. What happens then?
Answer: You can certainly do things that way. However, the strong scaling of pre-5E AC and attack bonus means that past the low levels, you risk a combination of grind and swinginess. PCs will only hit the monsters when they roll in the upper teens, but when they do, the monsters will explode. Conversely, the monsters will be trying to nibble the PCs to death with accurate but low-damage attacks.
Furthermore, you won't have any consistency in encounter power level. PCs operate on a very different power curve; the extreme changes in accuracy will have unpredictable effects; monsters with spells and spell-like abilities will be bizarre hybrids of 1E monsters with 5E spells; depending on how you handle DCs, monster "save-or-die" abilities may be phenomenally lethal. An encounter which was an easy fight in the original edition may turn out to be brutal in 5E. The next encounter, which used to be a serious threat, may prove a cakewalk.
Question 3: What if I use the adventure as written, but swap in 5E stats for the monsters?
Answer: Now we're talking. This is almost certainly faster than trying to apply a conversion template, and combat will run much smoother. You still won't have consistency in encounter power level, but that's okay, this is an old-school adventure, right? Your players shouldn't be expecting you to serve up nice safe level-appropriate encounters. If they get in over their heads, it's their own damn fault for picking fights they couldn't win.
You will still have to figure out how to convert custom monsters and classed NPCs, though. It will probably work best to re-create them from scratch using 5E's monster creation guidelines.
Question 4: I want to make sure the encounter power levels line up with the original edition. How can I do that?
Answer: Let's not kid ourselves: This is going to take work. First you have to figure out how difficult the encounters were in the original, for a party the same level as yours. This may be the hardest part of the whole thing! Assuming you can find a way to do that, you can then use 5E's tools for calculating threat levels and calibrate the encounters to match that difficulty.
When possible, it will be easier to adjust the number of monsters rather than changing their stats. Sometimes, though, you'll have situations where that isn't feasible. Changing "three orcs" to "six orcs" may not affect the adventure too much, but changing "one red dragon" to "two red dragons" probably will--and if you need to go the other way, you can't very well change "one red dragon" to "half a red dragon!" I'm sure 5E will have guidelines for customizing monsters, though.
As I said earlier, were I running an old-school adventure, I'd probably use a mix of the third and fourth approaches. For most encounters, I would just use the appropriate 5E stats from the Monster Manual, and let the players worry about gauging the encounter's power level. For the "boss fights," I would put in the work to calibrate the difficulty.
