Exploration is generally handled in a much more free-form manner
In some games, the shift from exploration or other non-combat conflicts to combat is barely noticeable. But 4e has very strict rules for its combat, that's the game's strongest point. In a way, 4e combat is its own separate game which gets initiated from time to time.
When not in combat, players can simply describe what they're doing. There's no initiative tracking or accounting for movement speed. It usually doesn't matter if it takes them 6 or 12 seconds, a minute or five to get from one point to another. It's enough for players to declare they'd like to get there.
So how do you adjudicate this?
- Give everyone a chance to provide input. Often, it's enough for one player to make a declaration and for others to nod. At other times you may wish to confirm with them what their characters are doing at this moment before something happens to change the situation. Though sometimes you may wish for them to have acted in a particular manner, for instance to have walked into your devious trap blindly, avoid deciding for them what they do.
- They don't need your permission for every little thing. Can they reasonably do what they're describing? Great, they do it. If you think there's some difficulty in achieving their goal, let them know, and ask for a skill check if they wish to continue. The players should have some idea of what the outcomes of passing or failing are, in order to make their decision meaningful. If it's a particularly complex thing they're trying to achieve, you may wish to make it a skill challenge.
- Put yourself in the characters' shoes. Describe what they see. Think of where they are, and what's happening around them. If there's something interesting in their vicinity, let them know, don't wait for them to guess.
This should answer your first two questions. With the third one, you've hit a problem dungeoneering has on the head. Yes, it is incredibly boring to have to declare you search every step for traps, rolling lots of dice for no benefit. This is partially solved by passive perception - remember to use that value to see if PCs simply notice a trap.
Overall, though, it's simply not interesting to have a trap whose sole challenge is in rolling a die to spot it. DMG has advice on using traps as part of a combat encounter, which is the 4e way.