[RPG] Silence spell and when to roll saving throw

Usually, yes. Silence specifically prevents sound, not all vibration. (Remember that heat is a vibration as well - a complete lack of vibration would indicate absolute zero, and probably herald the end of times.) It prevents only those vibrations that occur in ranges that can be heard. This stops most vibration-based direct attack spells (sound lance) and some vibration-based area-effect attack spells (soundburst), specifically those with the [sonic] keyword. At the DM/GM's discretion, it could stop or weaken some attacks besides spells that are a form of vibrative force if the judge rules that they're primarily or entirely composed of sound waves, but this should probably be done only if the supporting description of that attack describes it as being sound or sonic.

Magic is oddly arbitrary at times, but in any world where feather fall exists, we should be prepared for magic to interact strangely with physics.

It should have no effect on spells or abilities that use any other vibration frequency, although it likely has incidental effects on how the characters experience those spells - for example, a silent earthquake or meteor swarm is likely a very surreal experience. It can still be felt, and objects (and structures!) can be seen falling and smashing, but none of it can be heard - this would be strange enough for us in the modern day, just watching earthquake footage on YouTube with the sound muted; having it actually occurring around you with all the volume of a sensory deprivation chamber has got to be mind-bending.

To directly answer your examples:

Could they still be damaged by an Earthquake (PHB) or similar spell like Wrack Earth (PHB2)?

Definitely. Spells specifically let you know if they use sound in their effect by way of the [sonic] keyword.

How about the Shockwave spell? (Sharn: City of Towers)

I don't have this one, but assuming the spell has no [sonic] keyword, definitely. (If it does, clearly, it should be prevented.)

What about waves dropping heavy objects (for instance a boulder into a pond)?

Only if the sound itself is enough to cause damage. The water displaced by the boulder retains all its momentum as it enters the silence spell - that will hit at full effect. (Generally, if something hit something else hard enough to produce damaging sound waves, the actual effect of the hit is probably enough to make the sound damage negligible. For something crashing to earth, if it hits hard enough that the sound itself could damage a human standing 100 feet, the explosive impact and clouds of debris are probably going to kill that human.

What about weapons that cause concussive force types of damage?

Besides ones like the sound lance spell or a thundering greatsword or a sonic pistol, these should be unaffected by a silence spell - except the part where the opponent can't hear the weapons being used, so it's less likely to send up alarms, especially if the silenced combatant is in melee so the shouts and cries of his enemies can't be heard.

Interesting! On a similar note, would the same types of rules/ ideas apply to a deafened character? (or does a character need to be able to hear in order to be damaged by sound/ sonic attacks)

This might require a bit more case-by-case evaluation, but generally, a deafened character is only protected against [sonic] effects if they are also [mind-affecting] and/or [language-dependent]. A sound lance or sonic rifle will rip a deaf person's body apart and he won't even know what's happening to him, since he can't hear the cacophonous roar of a discharged physical sonic weapon. But spells like command and suggestion will completely fail to affect him, because they require him to hear and understand the caster's spoken words. A deafened character is very difficult to sway through normal Diplomacy or Bluff as well as many forms of magical persuasion short of outright domination. Non-combat uses of magical persuasion might be able to be rendered as a written message, depending on the spell; likewise, I've personally ruled that a deaf character was affected by command because they'd spent a Linguistics choice to gain the ability to read lips, and the conditions were met for them to automatically do so without making a Perception check. (Had they been further away or the caster not directly facing the deaf character speaking his one-word command slowly, I'd have permitted the deaf character to choose to fail their Perception check - in fact, I'd done so twice already that fight before the bad guy figured out why his spells weren't taking hold.)

No, casting Compelled Duel would end the Sanctuary effect on you.

As you quote:

If the warded creature makes an attack or casts a spell that affects an enemy creature, this spell ends.

Compelled Duel is a spell which affects an enemy creature, being cast by the warded creature (you), so it causes the sanctuary effect to end.

You can do it the other way around though

If you cast Compelled Duel and then have Sanctuary cast on you, the effects can overlap. A target successfully affected by Compelled Duel would have disadvantage on attacking anyone but you, but would be forced to make a wisdom save before they could attack you.

Note that the Wisdom save that Compelled Duel mentions happens only once, when you actually cast the spell on them. They either succeed, and therefore shrug off the effect and can act as normal, or they fail, and thus suffer disadvantage on attacks against other targets for the duration of the spell. The subsequent wisdom saves the spell may cause are only to do with the affected creature's movement - it still has disadvantage on attacks even if it successfully manages to move away from you (though you must remember to chase after it yourself if you don't want the effect to end).