Brand new and updated for 5th edition! Woohoo!
This post contains many frequently asked questions I've seen here many many times. They are divided into various categories. Page references are all from the main rulebook, unless stated. Some answers don't have page references - because sometimes the question assumes a rule exists, when in reality it does not... and it's quite hard to provide references to non-existent rules.
The answers below are not my opinions, they are GW rules. Where the matter is unclear, I have said so. Got any more you think should be added to this post? Let me know via PM and I'll add it in.

Also please PM me if you know I'm wrong (don't PM me if you only
think I'm wrong... check your rulebook first!).
These are 5th edition rules (though some are the same as 4th edition).
Shooting
What happens if a twin-linked/master-crafted weapon gets hot? Does it still injure the user even if I re-roll it?
No - if you re-roll any dice roll, the original result is ignored completely, and it is only the new result that counts. So if you rolled a 1 to hit when using (say) a master-crafted plasma pistol, and then re-rolled and got a 4, the original result of 1 is completely ignored.
(p31)
Can I re-roll a re-roll?
No, you can only ever re-roll a dice once; the first result is completely ignored and the second result stands.
(p2)
Can you shoot once with Rapid Fire weapons and still assault?
No. If you shoot at all with Rapid Fire weapons, or heavy weapons, you cannot assault afterwards. If you fire assault weapons, you can then assault afterwards. The exception to this rule is units that posess the
Relentless universal special rule.
(p28/29)
Can I fire at unit A then assault unit B?
No. You can never assault a different unit than you fired at, not even
Relentless units, and not even if you completely destroyed the first unit by firing. There are two exceptions to this however: (1) you must declare any assault the unit you shot at, but you can contact multiple units in an assault, including ones you did not shoot at -
p34, and (2) if you destroy a transport by shooting, you can assault the unit that disembarked from it.
p67
Do you get cover saves from blast/Ordnance weapons?
Yes. Note, however, that for barrage weapons, cover is counted from the centre of the template.
(p32)
If a unit suffers casualties from shooting, can you take off models that are not within the range of the weapon? Or with blast weapons, do I have to take casualties from under the blast marker?
Any model in the target unit can be removed as a casualty, even those not in Line of Sight or not in range.
(p24)
You can always see skimmers; right? They fly, after all.
No, they follow the same Line of Sight rules as every other vehicle.
(Skimmers, p71)
Can a unit in a transport shoot (from a fire point) at a different unit that the transport vehicle shoots at?
Yes, since the vehicle and the transported unit are two separate units.
Combat
Can you measure range before choosing a shooting/assaulting target?
No.
(Shooting p15, Assaulting p33)
If you fire assault weapons/pistols and then charge, do you lose the +1 attack bonus for charging?
No, you never lose this whatever you do before charging.
What happens if you consolidate into another enemy unit?
You can't - you must stay 1" away.
(p40 )
Are tanks ever in close combat?
No, neither tanks nor their attackers count as being engaged - so both may move away in subsequent turns, and both may be fired on. Any vehicle with a Weapon Skill attribute does count as engaged though, so neither it nor it's attackers can move away or be fired on.
(p63, Walkers p73)
Can Independent Characters single out models within a unit in combat?
No, and neither can Monstrous Creatures. However, they can of course choose to attack different units, and Independent Characters who have joined units can be singled out in this way - since for the duration of a combat (i.e. whilst attacks are being made) ICs count as a separate single-model unit.
(p49)
If a Walker is shaken and therefore not able to shoot, can it still assault?
Yes.
Morale
Can a unit that is falling back be tank shocked? If they can what happens?
Yes, they can be tank-shocked, but remember that a falling-back unit will automatically fail any Morale test it may be required to make (Regroup tests excepting). Remember that a Tank Shock will make them fall back immediately - so you can Tank Shock fleeing troops to make them fall back faster, and to prevent them Regrouping (since your tank is an enemy that is probably within 6").
(p46)
Psychic Powers
Can a psychic hood nullify any psychic power?
Since the test for it is taken "after an enemy model ... passes a Psychic test", it cannot be used against powers that do not require a Psychic test to function.
(Space Marine Codex, p 56)