I'd suggest you read the rules on the Perception skill, from the PHB.
1) Definitely - the exact map with all secret doors, etc, is yours; the players should only know what they've discovered. Old-school D&D actually forced the players to map, with one player sitting there with graph paper.
2) For most encounters, yes, the minis for the opponents and party should all be visible on the map from the start. However, you can make encounters more interesting by having creatures join in the 2nd round or later .. one classic example is the party coming to a T intersection, with some bad guys down the right-hand T ... (but more bad guys just out of sight down the left-hand T). The melee defenders rush to the right, and then the hidden bad guys come around the corner from the left.
For this case, I'll typically have the players make a Perception check to see if they notice the reinforcements as the reinforcements arrive at the edge of the battle. So, say the script calls for reinforcements to enter in Round 2. A failed Perception results in the reinforcements getting placed on the map during Round 2, on their initiative, and acting immediately. A successful Perception results in the reinforcements getting placed on the map on their initiative during Round 1.
3) Heck. No.
4) What can I say when someone asks: "Is there a secret in this room?".
A decent answer is:
"I don't know. Make a Perception check."
"I don't know. Do you want to spend ten minutes searching the room?"
A great answer would look like this:
"Just glancing around the room, you don't see anything - there's that desk on the north wall, and a chest in the corner, and there's a bookshelf along the south wall."
The scene is re-set; ask each player what they want to do,
then resolve what they do .. so maybe the rogue goes over to see if there's a trap on the chest, the wizard might look at the bookshelf to see if there are any good ritual books, the ranger searches the desk, while the fighter listens for approaching monsters.
The rogue has a Perception check to see if he notices the poison needle trap on the chest, which should have a DC to discover.
The wizard has a Perception check to see if he notices the false book which opens the secret door behind the bookshelf.
The ranger has a Perception check to see if he finds the ring hidden in the false compartment of one of the desk drawers.
The monsters sneaking up on the party make a Stealth check against the Passive Perception of the three active characters; since the fighter told me he was actively listening for approaching monsters, I let him make a Perception roll against the monsters' Stealth.