1) Yes, she can; and yes, she can.
2) Activating this is a standard action. So you can 'hold' until someone has spoken. However, you won't be able to read the truth from that.
3) Is a lance made primarily of steel?
5) Yes.
7) Huh? I don't follow your intention. Are you looking for Ride-by Attack?
8) Your mount could probably use sidestep if attacked. You, as a rider... not so much.
8. I suppose you could, but that would dismount you
7. You ARE looking for ride by attack.
6. There is nothing in the text that grants an additional +2 for Bull Rushing while charging. You gain the +2 from charge, thats it.
1) Can a summoned succubus give forever the +2 profane bonus to a PC? Or, when she returns in it's plane, can remove it?
She can remove Profane Gift as a free action. I see no restriction on from which plane. I would (house?) rule that it happens automatically when the summon spell ends.
2) Can an inquisitor activate discern lies immediately after someone has spoken?
Discern Lies (Sp): "Activating this ability is an immediate action."
An immediate action can be performed at any time - even if it's not your turn
3) Can be forged a lance in adamant
I think lances are generally made of wood. You could maybe get an adamantine-tipped lance or something. Ask your DM.
4) Is a natural 1 on concentration check a failure?
No. Natural 1 is failure on attacks, saving throws, and a few other things.
5) I'm casting summon eidolon defensively. Then I get hit in battle. I have to make another roll of concentration for the damage?
Yes. Casting Defensively is only to not provoke. If you take damage you still must concentrate normally. Getting hit, taking recurring damage, etc.
6) If i charge, I get +4 to bull rush (+2 charge, +2 charge to bull rush)?
Ask your DM. I think it's only a +2, since charge grants you a +2 on the attack made at the end of the charge, but Bull Rush replaces that attack. So the +2 to bull rush grants you the same bonus a charge would normally give you. Others may view it differently.
7) Can i use spring attack for make a charge in my mount?
No. Spring Attack is a full-round action where you move up to your speed. If your mount is spring attacking, it can't also be charging. If it's charging, it can't spring attack. You don't do either because you're not moving, the mount is.
8) Can i use sidestep from my mount? Who of the two must be attacked?
If the mount has Sidestep and it gets attacked and missed, it could use it. If you have Sidestep, I think you would need to dismount in order to use it after you were attacked.
I'm not real clear on mounted combat though, so hopefully someone else can come along and correct me if I'm wrong.
9) Can an eidolon with basic magic evolution have the arcane strike feat?
Arcane Strike: "Prerequisite: Ability to cast arcane spells."
Basic Magic (Sp): "An eidolon learns to cast a basic spell as a spell-like ability."
A spell-like ability is not the same as casting arcane spells.
Discern Lies (Sp): "Activating this ability is an immediate action."
My bad. I admit i stopped paying attention after the phrase 'as per the spell'
Still, if someone has already spoken, your ability kicks in too late.
With Ride by Attack i can't go away an opponet and charge it the same round. Or not?
Ride-By Attack (Combat): "When you are mounted and use the charge action, you may move and attack as if with a standard charge and then move again (continuing the straight line of the charge). Your total movement for the round can't exceed double your mounted speed. You and your mount do not provoke an attack of opportunity from the opponent that you attack."
So if "go away an opponet" means continue moving past the opponent while continuing the straight line of the charge, then you can. If it means move away from the opponent first, then charge him (or charge around a corner, etc.) then you can't.