Hey there, I was recently in your situation.
I was new to DnD and DM for the first time so I hope you can benefit from my advice.
1. My players went right to the Cragmaw Hideout after the ambush and we played until that part was completed.
They defeated Klarg after talking to Yeemik and were able to free Sildar.
We stopped with them on the road to Phandalin with a cart full of treasure and found supplies.
This took about 5 and a half hours with a few small breaks in between.
2. I can't remember exactly how I handled it but I remember the rogue scouted ahead.
I think she made a Stealth check which I checked against the passive perception of the enemies.
Since she didn't have darkvision I told her she only had a bit of light of the low burning fire and saw a few figures around it.
Since there was no sign of Sildar or Gundren and there were alot of enemies she headed back and they decided to take another path.
3. Here's what happened in my playthrough.
I'm copying it from a Reddit topic where I posted it before.
The flood event also depends on the goblin on the bridge that is hiding but not really paying attention.
In my playthrough the following happened.
A player with Darkvision sneaked ahead so I let him roll Stealth.
His Stealth roll was high enough to beat the Goblin (passive Perception of 9) so he wasn't detected.
The player asked what he saw so I let him roll Perception.
His Perception roll wasn't higher than the Stealth roll of the Goblin I made so he only saw the bridge above the water but not that someone was on it.
Another players joined him and again his Stealth roll was high enough to avoid detection.
I also let him roll Perception and he beat the Stealth roll of the Goblin so I told him that he saw a Goblin hiding really well against the wall.
They tried to take him out so I gave them both advantage on their ranged attack rolls.
They failed to kill him though so after the surprise round I let them roll Initiative.
The Goblin didn't act first but since the players missed their attacks he ran away to the Pool chamber and they heard him screaming that there were intruders.
All the players moved closer and were discussing what to do.
The turn after I told them they heard knocking.
They moved around the corner and saw what the Goblins were doing.
Since they determined they couldn't take them out in time they decided to run back to the side corrider to hide.
One of them only could get to the opposite side of the river so he stood there while the flood was coming.
One player threw him a rope to keep him in place if needed. (I would have given him advantage on the Strength saving throw if he failed his Dex saving throw)
Everyone was watching in excitement when I told him to make a Dexterity saving throw to jump out of the way in time.
He rolled a natural 20! He made it safely to the other side
4. I would try out the Theatre of The Mind (ToTM) approach which is the default for 5e.
You describe the surroundings, everyone's positions, distances etc without going into too much detail.
You (and your players) might be surprised about how easy and flexible this is.
It also uses more of your imagination.
You can use it for the ambush part and see if you like it.
Keep in mind that it is important to repeat each combat round where everyone is to keep things alligned.
For the Cragmaw Hideout I wanted to use the map in the DM booklet.
Off course that one has spoilers on it so it can't be used.
I bought the spoiler-free Player version from the artists website on
http://mikeschley.zenfolio.com/p856083253
It doesn't cost much and is a nice way to support a great artist (who proclaimed he needs this income to make a decent living)
I printed it on A3 paper and cut it in different parts.
As the players explored I put more and more on the table.
This seemed to work well.
I didn't use miniatures or exact distances though.
I used it to indicate everyone's position.
In doubt I tried to rule in the PC's favor.
I have to say recognise myself in you.
I had the same trouble figuring out the math behind the numbers (your combat thread) and also wanted to be super prepared for my first session.
In the end it went really well for me and I'm now preparing the 2nd session.
I wish the same to you and I hope this helps
Let us know how it went, I'm very interested.