Can you continue to be not 'clearly visible' while moving and remain Hidden?
Yes, unless you are travelling for long distances at a normal or fast pace.
Can you become 'clearly visible' while moving and remain Hidden?
Yes, if the circumstance allows it, but usually not when you approach someone in combat.
Movement matters for hiding in two general cases
When you move over a distance measured in days, hours, or miles, this paragraph in Activity while Traveling applies:
While traveling at a slow pace, the characters can move stealthily. As long as they're not in the open, they can try to surprise or sneak by other creatures they encounter. See the rules for hiding in chapter 7.
While you are in combat, this paragraph of the Hiding sidebar applies:
In combat, most creatures stay alert for signs of danger all around, so if you come out of hiding and approach a creature, it usually sees you. However, under certain circumstances, the Dungeon Master might allow you to stay hidden as you approach a creature that is distracted, allowing you to gain advantage on an attack before you are seen.
There are also a handful of specific features and traits which tie movement and stealth together to varying degrees, such as the Ranger's Hide in Plain Sight feature or the Gargoyle's False Appearance trait.
You can't be both clearly seen and hidden from a creature
The Hiding sidebar states:
You can't hide from a creature that can see you clearly, and if you make noise (such as shouting a warning or knocking over a vase), you give away your position.
This applies throughout the whole time your are hidden, not just the moment when you begin to hide. Moreover, the sidebar states:
Ultimately, the DM decides when circumstances are appropriate for hiding.
Which implies a change in circumstances can cause you to become unhidden. This is consistent with the Ability Checks chapter, which states:
An ability check tests a character's or monster's innate talent and training in an effort to overcome a challenge. The DM calls for an ability check when a character or monster attempts an action (other than an attack) that has a chance of failure. When the outcome is uncertain, the dice determine the results.
If the outcome is certain, then dice do not determine the result. If the circumstance changes such that you are clearly seen, then by definition the outcome of hiding is certain failure and the stealth check no longer determines the outcome.
That said, whether being clearly visible means you are clearly seen depends on the circumstances. For instance, if you are out in the open and approaching a sleeping creature, you are clearly visible, but it's uncertain whether that creature wakes up and sees you, so the stealth check determines the outcome.
Hiding works pretty much like you think it does
Most of the relevant rules are freely available online, here.
1. Do I roll a d20 + DEX (stealth) when I first want to hide, and note down the number?
That's basically it. Generally the check will be requested by the DM when the player informs them that they wish to hide, rather than rolled pre-emptively by the player.
Remember, no one can hide in plain sight - a DM might not let you attempt to take the hide action if there isn't anywhere for you to conceivably hide. A mean DM might let you take the action but tell you it automaticially fails, if you try to hide somewhere inappropriate.
However long you remain hidden, normally you won't need to reroll a new stealth check, unless something changes.
2. Then, for each turn that a creature is actively looking, do they roll a WIS (perception) for it and compare this against the written value?
Right again, Wisdom (perception) is the normal way to detect the presence of something hidden.
That said, a creature wouldn't usually be 'actively looking' for multiple turns. Certainly a lot of DMs wouldn't allow that. If a failed check can be easily rerolled on the next turn then it has no consequence. Many DMs would tell the PC that 'it looks like maybe there was no one there after all' if they fail to find someone. Carrying on actively looking under those circumstances could be considered a bit meta-gamey.
I might occasionally allow a PC to reroll a failed perception check, but only if they had a good in-character reason to believe they'd missed something the first time. I certainly wouldn't let them roll every round. Whether you do is up to you - bear in mind this guidance from the DMG:
Sometimes a character fails an ability check and wants to try again. In some cases, a character is free to do so; the only real cost is the time it takes.[...] In other cases, failing an ability check makes it impossible to make the same check to do the same thing again.
If a PC wants to make a particular check with extreme care, taking more time in game to make sure they haven't missed something, then we could use passive checks which 'can represent the average result for a task done repeatedly'. However, that won't help them much as their passive perception should probably already be treated as the floor to their active perception rolls. That being the case, I would let them take 10 minutes and reroll instead.
3. If a creature isn't actively looking, do I instead compare its "WIS (perception) score + 10" to the written value?
Three in a row! That's the right way to calculate passive perception. Sometimes people notice things that they aren't actively looking for - just like real life.
4. What happens when a "check" results in a draw. Who "wins" here?
When two characters make competing ability checks this is what's called a Contest or an opposed check.
If the contest results in a tie, the situation remains the same as it was before the contest. Thus, one contestant might win the contest by default.
What does it mean when it says that 'the situation remains the same'? Luckily there's a couple of handy examples...
If two characters tie in a contest to snatch a ring off the floor, neither character grabs it. In a contest between a monster trying to open a door and an adventurer trying to keep the door closed, a tie means that the door remains shut.
So, in a contest where one creature is trying to hide and the other is trying to see them - nothing changes - the creature remains visible.
Best Answer
To answer your question directly, yes. You are correct.
From what I understand though, you are confused about a) the "DC" of your hide check, and b) what happens after you are spotted, and targeted.
Well firstly, a creature's Passive Perception is calculated by adding 10 to their Wisdom Modifier. This means that a creature would need a Wis Score of 16 to spot you with passive perception. This is unlikely. However, the GM may have decided to give them advantage, since they got the drop on you. This means that they get an additional +5 to their Passive Perception score, which means they only requires a Wisdom of 8 in total to still spot you. (10 + 5 - 2 = 13)
Secondly, if they do not spot you, and need to search for you, even if only one succeeds, the gig is up. You've been spotted, and your position has been given away by the monster hacking at you. Even a monster with only 8 Wisdom would still be able to figure out that there's something worth attacking, if one of his friends was having a go at something they couldn't see.