Up to D&D
Herein I address such eternal questions as:
Sometimes the answers are directly out of the book or the other book. Sometimes they're my own way of doing things. Seldom are they my own inventions.
Anybody can hide. Hiding is an all or nothing thing. Either you're hidden or you're in view.
Thieves, however, have a chance to hide in shadows. A thief generally prefers to hide rather than to hide in shadows. It's nice to have a chance to hide in shadows when you need it, though.
Halflings have a similar ability, having a 2 in 6 chance of hiding in shadows. They also have a 90% chance of hiding in woods or underbrush. (p.B10)
Note that to hide in shadows, the thief cannot be moving, silently or otherwise. Hide in shadows & move silently are mutually exclusive. There is no double jeopardy.
Anybody can move quietly. I like to use a variation on the listening at doors mechanic (p.B21) to determine whether a character moving quietly is heard.
Demihumans have a base 2 in 6 chance of hearing noise. The chance for thieves depends upon their level. For everyone else it's 1 in 6. (The DM will choose a base chance for each monster.) If the sneaking character is wearing metal armor or walking on a particularly noisy surface, increase the chance of being heard by 1 in 6. (i.e. 1 in 6 becomes 2 in 6. 2 in 6 becomes 3 in 6.) If multiple or extreme factors are involved, the DM may increase the chance further.
Increases beyond 5 in 6 should go to 7 in 8, then 9 in 10, then 11 in 12.
Thieves, however, have a chance to move silently. If successful, there is no chance for the thief to be heard. If the thief fails to move silently, they should still be considered to be moving quietly.
Note that if he is moving silently, a thief is not hiding, in shadows or otherwise. Move silently & hide in shadows are mutually exclusive. There is no double jeopardy.
Note: The surprise mechanic can also be use to cover stealth. There is precedence for this in module X4:
[the characters] will first have to dispatch the guard ... Since he is not watchful, the characters will be able to surprise him on a 1-3 if they creep up on him.
(...from Mike's observation on DF)
Anybody can search for traps. Dwarfs have a 2 in 6 chance. Everyone else has a 1 in 6 chance. (p.B22)
Thieves use the 1 in 6 chance until 3rd level when their F&RT skill becomes more favorable. The 1st & 2nd level thieves, however, must use their less favorable F&RT skill when attempting to disarm a trap.
Anybody can hear noise. Demihumans (dwarfs, elves, & halflings) have a 2 in 6 chance. Everyone else has a 1 in 6 chance. (p.B21)
Anybody can climb. Generally, the DM can simply declare something climbable or inclimbable. In stressful or unusual situations, the DM may call for an appropriate ability check.
Thieves, however, have a chance to climb sheer surfaces.
These are thief only skills.
A non-thief might be able to learn these abilities by spending appropriate costs (time, money, XP, or whatever). The skill would not be tied to the character's level, but would require additional training to be improved.
I think it is worth mentioning that I used to think the names of the thief skills were hyperbole for flavor. I read “hide in shadows”, “move silently”, & “climb sheer surfaces” as simply “hide”, “sneak”, & “climb”. I even defended this position steadfastly. When I finally saw things from the other point of view, however, I liked it.