He doesn't just insantly know all of the monsters stats. Theres a set DC to find out what it is and a tidbit of information. Every 5 after is another tidbit(what a tidbit is, is up to you). 10+ CR is the usual. Telling you to just show you the whole things character sheet sounds rude and breaks immersion to the nth level.
He gets to make them when he ask and as a reaction. As far as he knows the golem is a statue until it moves or he examines it. When it starts moving he can roll one as a reaction, on his turn. I think thats how it works anyway... Its how I've done it. I know that much.
You might want to have a talk with them to decide some things before the next session. When things get up in arms or people are demanding things, it can make people uncomfortable. Kindness and doing things politely is a big thing sometimes. Think that might help?
If a magic item, such as your statue/golem is disguised, it is quite fair to make a character succeed on a Perception check, before even attempting a knowledge roll. Unless this player goes about suspecting every statue he sees is a golem, I cannot see how he can argue with that.
Here's how we've been doing it for a long time in my group:
Outside of combat, there's no limit to knowledge checks as long as the player (not the GM) requests them. Other players can assist on an out-of-combat knowledge check since it's assumed their assistance is in the form of talking about the issue/monster/whatever at the time. However, any player that assists can not make separate knowledge checks on the exact same thing, since their assist roll indicates they're already adding the information they know to the check. Any players that choose not to assist can roll a knowledge check as usual.
In combat, we've been doing 1 knowledge check per turn on the player's turn as a free action. The player decides which check to use, typically based on the description of the monster. If the player doesn't get the info he wants (I.E., he chose Knowledge: Arcane instead of Knowledge: Dungeoneering), he can then use a swift action to get another knowledge check of his choice. If he still isn't satisfied, he can then use a move action to get another knowledge check of his choice. If that STILL doesn't work, he can then use his standard action to get a 4th knowledge check of his choice.
Our in-combat knowledge check system means the player can spend a full round action for 4 knowledge checks, which represents the fact that he's wracking his brain trying to recall the specific details he needs. Most of the time our players will use a free action and possibly a swift action, so they can still take a move and a standard action. It also caps it to 4 knowledge checks per turn, which seems a fair tradeoff given the increased chance at success and requiring a full round action to roll the checks (and it isn't ridiculous enough to allow checks on ALL knowledge skills, just the 4 knowledges that the player thinks are the most relevant).
Our system's been working really nicely at my game table, and there have not been any complaints from players or GMs about it since we adopted it 2-3 years ago.
You're also absolutely right, it's up to the player (not the GM) to ask for knowledge checks. If the player looks like he's trying to do something that would require a knowledge check or if a knowledge check seems appropriate, there's nothing wrong with suggesting that the player roll the check.... sometimes a module will say "Knowledge Check XYZ will reveal this", and there may be nothing in the room's description to hint that a knowledge check would help. For instance, if Player Z is a ranger with constructs as his favored enemy for the last 6 levels, and he goes into a room with an inert golem that looks like a statue, I would tell him he can make a free Knowledge: Arcana check if he wants. Don't tell him why he gets to make the check or what prompted the check, just let him roll it, see what the result is, and clue him in with whatever details you see fit: a result of 15 might just be "something feels wrong in this room, but you're not sure what", but a result of 25 might be "you're pretty sure that statue in the corner is a stone golem".
EDIT: If the player tries to argue with you, compare Knowledge to the other skills. You don't tell players when to be stealthy (and use the Stealth skill), you don't tell the players when to appraise items, or when to lie to guards, or when to unlock doors or pickpocket nobles, or when to use a wand with Use Magic Device. In very, very limited situations you require Swim checks, Acrobatics checks (for balance), and Climb checks. But 97% of the time it's up to the player to decide what his character does, and that includes knowledge checks. If a player chooses not to use his +37 Knowledge: Dungeoneering check on something, the character may simply not think the description of the room is strange enough to take notice of.
You can also ask if anyone wants to make any knowledge checks after they enter any room. Read the description, ask if anyone wants to make 1 knowledge check of their choice, and then continue play. That way you give them a chance, and if they choose not to make a check (or any additional checks), then it's on them and not you as the GM.
Knowledge Checks...
1 He started to yell and throw a fit that I should have told him to make a arcane knowledge check the moment the creature came into view not roll it in secret.2. He also demanded that I show him all the creatures stats, Exactly how much health it had and all of its weaknesses... which takes a lot of the fun out of it personally.
3. I'd much prefer that when trying to figure something out they choose to apply there knowledges of their own accord and then make there own rolls.
It takes out half the effort if i tell them exactly which knowledge to apply4. Wait until it was his turn to apply his skills... this angered my players as they demanded that since knowledge checks and talking are free actions you can do them at any point during the battle whether its your turn or not.
1. First off, yelling and throwing a fit is never acceptable, even if you don't see eye to eye with your GM. I recommend having a conversation with this player about appropriate behavior at the table. Next, in the scenario described, I wouldn't give the player a knowledge check unless they investigated the statues specifically and in detail, as in going right up to it and peering at each joint etc. Not all constructs look exactly alike and many look just like statues. A perception check might be appropriate. Perceptions checks are often rolled in secret anyway.
2. Sounds like a very demanding player. Several people responded with the knowledge check rules on only giving out certain amounts of info, which is what he should get. Personally, I offer the player a choice for the first tidbit, details on either a creatures attacks, specials, or defenses. Usually I give out general traits like undead or elemental traits with the creature if they ask. I also give out a lot of extra info beyond the normal amount of tidbits if they exceed the DC by 20. Kind of like a critical knowledge roll, though this is extremely rare to the point I can't remember the last time it happened and should never be given out just because someone rolls a nat 20.
3. I disagree with you on this one; knowledge checks are still plenty difficult. When you usually only get 1 or two tidbits of info with most succesful checks anyway, and on a low roll (only bards can take 10) you can still fail it in the first place, its annoying to have your own character fail to think correctly. I don't accidently think about the borders of my country when trying to remember what I had for lunch last night for example, or when faced with a car I don't think hard on what kind of building it is. Pathfinder obscures this issue a bit by dividing different creatures among different knowledge types for game balance. Personally, I see it more as a character thinking about all the different creature types he or she knows trying to recognize it more like a knowledge: creatures, rather than specific subgroups. Knowledge creatures, of course, would be totally unbalanced in game and can't exist, so I think it is fair to let players roll the appropriate knowledge as long as they have it.
It can also be difficult to distinguish what type a creature is from it's description.
For instance, in a recent game I played in, we had two skeleton trolls and two dog like creatures with no skin, only exposed musculature that had multiple heads. Well, the skeletons reinforced the idea to me that the dogs were some sort of variant zombie or undead, so I rolled knowledge religion. Turned out they were cerberi, so the appropriate knowledge was planar. My GM and I don't see eye to eye on this issue, so I was unable to try to recognize the creature with Know: planar until the next round. Of course, the next round I rolled a 2 on my knowledge check and failed it anyway.
Laardrym wrote in an interesting compromise to these two styles above, using a swift action to get another check for a different knowledge type, escalating to a move then standard after that. I might suggest that to my own GM as well, I'd like it better than waiting a round to try a different check and he'd probably like me to stop teleporting away from AoO's with my swift actions.
4. You were totally in the right here, Knowledge checks are definitely not immediate actions. I would point out, however, that if the players are not in initiative, they should be able to make the check at almost any time they call it.