You can find a description of each condition within the Codex which will tell you what rules pertain to it. For Daze specifically we have:
Daze (Condition Marker)
Creature is disoriented and/or blinded. Whenever this creature makes an attack, roll the effect die at the end of the Declare Attack Step. If the attack skips that step, then roll as soon as the attack starts. If the result is 7 or higher, the attack is resolved normally. If the result is 6 or less, the attack “misses” its target and is considered to have been avoided (even if the attack is Unavoidable). If it makes a Zone Attack, only check once: the entire attack either fails or succeeds. If a creature has more than one Daze on it, roll only one time to see if the creature misses. In addition, the Dazed creature suffers a -2 penalty to all Defense rolls for each Daze marker it has. All Daze markers are removed at the end of the creature’s Action Phase. Daze has a removal cost of 2. Conjurations cannot be Dazed.
This tells us that when dazed you must roll the effect die for EACH attack that the creature makes. If the die lands on a 6 or below, the attack fails. The creature also suffers a -2 penalty to Defense rolls for each Daze marker on it. Daze has no other effects and all Daze markers are removed at the end of the creature's action phase.
I'll now address your specific questions, incase that has not clarified the issues:
1) Yes, it may still act completely as normal. The daze description does not mention any limitations on what actions it may perform.
2) Yes, the condition marker comes off at the end of the creature's action phase regardless of what actions they did or did not take.
NOT at the end of the attack, mind you. If the creature makes multiple attacks, or makes an attack outside of it's action phase (such as with counterstrike), the Daze condition is not removed. In this way, it can affect multiple attacks.
3) Yes, the daze description does not mention limiting it's ability to hinder so it is still capable of doing so.
4) No, you never have to pay a removal cost unless a spell or effect tells you to pay it.
5) You would have to pay it if you used a spell or effect that told you you had to pay it. Such as with the Priestess's restore ability or the Wand of Healing.