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Is it always a third that "leads" to the root? For example, if we have GBD, it's B that leads to the C. Could it be also BDF that leads to C, or EGB? All of them contain B, a tone that leads to C. Do all of them lead to C? Can all of these B act as the leading tones or only the one in third (GBD)?

Second thing is about the half step thing. Can we resolve a half step lower, isntead of resolving by going higher? What I mean is can we resolve to Bb (going down) instead of C (going up) when we have B as a leading tone (again with GBD chord, for example)?

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You have more than one question wrapped up in your post.

First, the fact that other chords (in the key of C) have the B in them doesn't mean you will create the feeling of tension and relief when moving from one to the other. You example of EBG (the minor chord on the iii of the key) does not really create this feeling, there is movement but not "strong" movement. The GBD chord is the V (five) of the key and has a special relation to the CEG (the one) chord. The movement of GBD to CEG involves several intervals and it's the relation of all of them that creates a resolution. In classical theory books they would state that the "V has a natural tendency to want to move to the I". This doesn't mean that other movement is not acceptable, but none have that strong or natural tendency. As Tim points out there is the V7 chord, in the key of C this is GBDF. The movement of V7 --> I has 2 half step movements B-->C and F-->E which further strengthens the feeling of tension and resolution.

In your example of EGB --> CEG, iii --> I, the iii chord is a natural extension of the I as it creates the major seventh chord, CEGB. In fact, in Jazz the iii chord is sometimes used to substitute the I, which creates no real feel of movement other than changing inversion. Part of what creates the tension and release in the V-->I is the specific inversions used. We typically try to create close movement from one chord to the other. The fact is that not all V-->I movements have the same degree of tension-release, or resolution. So, you can create more (or less) resolution based on your choice of specific note movement from one chord to the other.

In response to your idea of moving backwards from C to B, yes you absolutely can create resolution that way but it may not work as well as B-->C in the key of C. If you are in C, the natural B chord is a diminished. Moving to a diminished chord from a major chord would create tension rather than remove tension. But in a string of chords one could keep moving from B(dim) to E7 to A(min) and there would eventually be resolution to the A(min).

The V chord is really what makes the resolution (in the classical sense) work. Along these lines Jazz musicians will frequently "fill in" progressions by adding the V7 chord before each chord of an existing progression, treating the original chords as temporary I chords. The device is really V7-->I rather than move a half step.

As a final comment to your question about moving using other half steps, there are only 2 half steps in the Major scale so your only options are 7-->8 and 4-->3. This is present in the V7-->I chord change. If you want to stay "in key" you don't have other options but you can always use chromatic notes to create more tension. A common progression is I--> I#(dim)-->ii-->ii#(dim)-->iii-->IV or some variant of this. It creates a slow, smooth, chromatic walk up from the I to the IV that contains some feel of resolution in it.

So, you can do what you want in a creative sense but there is a tested method to making this work.

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The third (essentially major third) of V is called the leading note/tone for that very reason. It leads. more often than not, and sounds like it should, if It doesn't (!) to the root note of I.

Often the 'tension/release' involves movement of a semitone - the smallest step in Western music. In the V(7)>I, named the perfect cadence, there's another semitone step. That's between the 7th of the V chord - in key C, the note F - which resolves nicely downwards to the defining third of C, the I chord.

There's also another downward resolution which involves the Vb9 chord, G b9 here. Note - not Gb9, aka F#9, a completely different chord. Sometimes it's written G7b9 to avoid ambiguity. The b9 part of G7 is Ab, which again has a semitone to move to the G note (5 of root C).

There's another interesting cadence tritone substitution, which involves a move from Db7 to C. Db7 is made up from Db, F, Ab and B, and you'll notice an F and a B in there. The two pivot notes which in key C are involved in G7! That chord also uses the leading note to get back to tonic, root I.

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Considering C major as the key, notes G B D represent chord G, which functions as the dominant V chord. As a dominant, tension is high and tends to go to a tonic function chord like C. Notes B D F is chord Bdim, that functions as the VII grade which also happens to be dominant. B is half-step from the tonic C, this might be the reason why these 2 chords tends to go back to C and release tension.

However, there is another chord that also has B, that is Em (E G B) which is grade III and works as mediant. The reason for not being classified as dominant might be because it shares 2 of the 3 notes with the tonic C chord (E and G). This is an example where a regular chord with B might not create much tension.

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As there's been no response to a couple of perfectly good answers, shall I try saying much the same thing in a different way?

Within the context of C major tonality, B is the leading note, with a tendency towards resolving on C. This tendency is reinforced by including it in a G major chord - G, the dominant note - also has a tendency towards C - the tonic. If we add an F - a tritone interval from B - forming a G7 chord, the tendency becomes even stronger as the F tends towards E, part of the tonic chord.

But yes, the B can go to other places. Music is full of both expected movements - 'resolutions' - and unexpected ones. Many of the unexpected ones will involve a move away from C major. For instance, G7 (B melody) could resolve to A major (C# melody).

Or we could take advantage of the symmetrical nature of the tritone to harmonise F-B not as part of a G7 chord but as part of Db7 (better call the B a Cb though!). F will become the leading note of Gb major.

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