[RPG] When exactly do I make a saving throw

Yes

From the Rules Compendium, p260 (emphasis mine):

Death Saving Throw: When an adventurer is dying, make a saving throw at the end of that character's turn each round. A death saving throw works just like any other saving throw, but the result determines how close the adventurer is to death.

If you get a bonus to all saving throws, that bonus does apply to death saves.

As you and your group correctly suppose, the silence spell works at its best when cast upon an area or an ally.
No SR applies and no ST is required in this case but the area is stationary or moves with the ally, so the intended target (most probably an enemy caster you're trying to silence) could still move outside of the area and cast his spells.

While there are means to prevent this escape (mostly by grappling or tripping the caster or by readying actions to follow him when he moves) there still is some utility in having a silence area that follows the enemy by itself. Of course, the enemy should be the caster you want to silence, not a random foe.
This is a better tactic for it's harder to avoid and/or does not keep one of your party busy with following or disabling the caster. As such, it's perfectly OK if it's harder to set up. The only drawback is that the enemy could also try to get to your casters and silence them as well.
But that casting method has a reason to exist and yes, it works just like you and your friends said.

Only the target of the spell (the centered creature), if any, is subject to ST and SR