You ask about cheat sheets, but your problem is actually "how to make the game flow well without long rule-checking breaks." That is what I will address.
First of all, what a great resource you have found. I have immediately printed it off to use tonight. That said, there is way too much information here - you almost don't need the books - just use the last page on combat as part of your "screen".
From the tone of your question, I assume that you think that stopping the game to look up rules is a bad thing. I mention this because this is very group specific - for some groups learning the game (and these rules are so new that everyone is learning) is part of the fun. I just make this point and leave it there.
Luckily for you there D&D is really easy, how to play is on page 6 of the Players Handbook:
- The DM describes the environment.
- The players describe what they want to do.
- The DM narrates the results of the adventurers’ actions.
Well, you can't be having a problem with No 2 because that's not the DM's problem, so we only need to look at 1 & 3.
1. The DM describes the environment.
You have to know what it is, luckily there are only 2 parts to this people and things.
Things are the easiest - you have to have read the module, preferably twice, and have reread the bit where the PCs are up to just before you start playing. So, describe the things (places, traps, objects etc.) and let the players tell you what they want to do with them.
People are a bit harder. The people in this module have bits of information scattered through the book. So I built a mind-map (contains spoilers for Lost Mine of Phandelver):
This shows how people are connected to things so I can look at this and know if they are talking to X then he is connected (in some way) with A, B, C & D - this is enough of a trigger (for me anyway) to allow me to remember what the NPC knows without having to thumb through the book.
When the PCs meet an NPC for the first time decide how to play them (e.g. timid, shy & high voice or blunt, gruff & swearwords) and WRITE THIS DOWN so you can be consistent the next time they meet. Generally, play to type BUT the NPCs the players remember are the ones that stand out ("You remember, the effeminate half-orc barbarian with the pink parasol").
3. The DM narrates the results of the adventurers’ actions.
Let me highlight the important bit: the DM narrates the results, not "the DM works it out" or "the player's debate with the DM" or "everyone spends 15 minutes with their head in the books" or even "somebody rolls a dice".
You're getting paid (or more probably not paid) to make a decision; make it!
This doesn't mean that you are arbitrary or unfair - level with your players, tell them that if they can't give you the rule inside 15 seconds you will make the decision and that will be binding until the end of that period of play. The rule, not the reference - believe what they say - most people don't cheat and even if they do, a ruling that is kind on the PCs now is likely to be kind on the monsters later. Set aside some time during or after the session to look up the rules no one knew together and use that rule going forward. This is a process called learning.
If we are dealing with combat, well, you now have your handy flow chart, that should cover 95% of things that can happen in combat.
Know the capabilities of the monsters the PCs are likely to fight - I photocopied the monster stats for this module and highlighted the key things (Goblins "Sneak", Bugbears are "Brutal") - this makes a much bigger difference to the way they play than things like hit points or armor class.
If its a non-combat situation (or a situation in combat that's not covered), then ask yourself the following question:
"Does anything of significance hinge on the player failing to do this?"
If the answer is "no" then they do it; if the answer is "yes" then decide if what they want to do is easy (DC10), medium (DC15) or hard (DC20) and if circumstances give them advantage or disadvantage - tell them the target and let them roll the dice. If you feel generous (I'm always very generous) give them 10 seconds to convince you that you're wrong or to change their mind about doing it.
Some things are impossible - you can't read a closed book, fly without wings or magic or hold your breath for an hour - if its impossible, say so.
For spells, make the PCs responsible for knowing the effects of their spells - you have enough to do.
Two final points:
- you don't get graded on this,
- make sure everyone has fun!