There are 6 combat states in 5e when you don't consider cover coming into play (and cover adds modifiers rather than affecting the advantage state, though full cover prevents targeting entirely):

• Able to target no adv/disadv
• Able to target adv
• Able to target disadv
• Guess no adv/disadv
• Guess adv
• Guess disadv

The question is how total concealment (such as from fog cloud) affects these states. To do this we need to address the two components of the states individually. Let's settle the question of advantage/disadvantage first.

A creature in a heavily obscured area effectively suffers from the blinded condition (see the appendix). (Player's Basic 65)

This leads us to the blinded condition:

• A blinded creature can’t see and automatically fails any ability check that requires sight.
• Attack rolls against the creature have advantage, and the creature’s attack rolls have disadvantage. (Player's Basic 105)

So here's the interesting thing, you both grant advantage, and have disadvantage if you are in a heavily obscured area. So if you attack a creature who is also in a heavily obscured area, you do so as a straight roll (because they are granting advantage and you have advantage). This may be dissatisfying (I know I find it such), and you may want to rule that you only grant advantage to characters who can see you (but that's a house rule).

So we know our advantage state in heavy concealment (attacking at disadvantage, being attacked at advantage, possibly cancelling each other out if both characters are in the fog). However, we don't know our targeting state. Can you be targetted directly or do you need to guess. For this we turn to the "attacking things you can't see" section of the rules:

When you attack a target that you can’t see, you have disadvantage on the attack roll. This is true whether you’re guessing the target’s location or you’re targeting a creature you can hear but not see. If the target isn’t in the location you targeted, you automatically miss, but the DM typically just says that the attack missed, not whether you guessed the target’s location correctly. (Player's Basic 73)

This is unfortunately unhelpful. When do you have to guess, and when do you know where your target is?

Unfortunately, the guidelines are not particularly good on this. It seems clear to me that hiding is intended to conceal your position (and it's heavily implied elsewhere in the rules, such as the rogue's Blindsense feature, and the final paragraph of the attacking things you can't see block), but it does not directly indicate that your position becomes unknown when you hide. There is this though:

You can’t hide from a creature that can see you, and if you make noise (such as shouting a warning or knocking over a vase), you give away your position. An invisible creature can’t be seen, so it can always try to hide. Signs of its passage might still be noticed, however, and it still has to stay quiet. (Player's Basic 60)

So it seems that hidden certainly obscures your position, and if you move, it should be unknown to your attackers (if you don't move, they'd be perfectly justified to attack the square you were last seen in, thus being able to attack you no problem).

The larger question is can you attack a creature who isn't hiding, but is totally obscured without guessing their location?

This is a question that is unfortunately left completely open by the rules. In general, I'd say that you need to make the stealth check to hide to keep quite and keep your movements under control, but that may be the 4e coming out more than it should in this system. I kind of hope that this gets more clarity in the upcoming DMG than we currently have, but I'm not super hopeful about it. Find something that works for you group and keep it consistent (or establish the rules you're going to operate under when these things come into play). One place we do get a hint is from the invisible condition:

An invisible creature is impossible to see without the aid of magic or a special sense. For the purpose of hiding, the creature is heavily obscured. The creature’s location can be detected by any noise it makes or any tracks it leaves. (Player's Basic 105)

This certainly seems to indicate that you can know an invisible or heavily obscured creature's location, but that it doesn't necessarily require a stealth check (again I'd personally require one to stay quiet)

To answer your final bullets:

1. Movement does not appear to be hindered by obscured conditions. If a DM wants to hinder movement they do so at their discretion.
2. By the book, if they aren't hidden (position known), the attack is made as a straight roll. The creature attacking has disadvantage on the attack because they can't see, but the creature being attacked grants advantage. This seems dumb to me, and I'd make the attack happen at disadvantage.
3. This attack is again made as a straight roll provided the creature isn't hidden (ie position known). Again since the attacker can't see the creature, they have disadvantage, and because the attacker can't be seen, the target has disadvantage.
4. Same story as above, you can't see your target, target can't see you, straight roll.

So, ultimately, this question comes down to one of whether or not you can target someone accurately if they are heavily obscured. This, ultimately comes down to whether you believe position is always known unless someone is hidden, or whether position can be obscured by another means beyond hidden. Unfortunately, the rules are not clear on this matter and it becomes squarely the domain of DM discretion or table level house rules.

Supporting this stance, we have Jeremy Crawford on the twitters:

Being hidden is the by-the-book way to conceal your position. The DM may decide that other methods can also conceal it.

So basically, it's hidden, or the whim of the DM.