Does this interaction allow me never to take damage?
Or, does Force Bubble's replacement effect have to happen to avoid me taking damage? (Making Solemnity not useful in this scenario)
In general, both, but Force Bubble's replacement effect won't have you put any counters on anything.
In general, if you control Force Bubble, Solemnity is on the battlefield, and you would be dealt damage, then nothing would happen instead — since you can't put any counters on any "artifacts, creatures, enchantments, or lands", including Force Bubble, you don't put any (C.R. 101.2, 101.3, 614.6).
Note that the only thing in this scenario that affects whether damage is dealt is Force Bubble's first ability. Solemnity neither prevents damage nor makes damage impossible.
If I control both of these cards, does that mean I will be able to avoid taking damage? I understand that damage was still being dealt. For example, if creatures attacking me have lifegain, their controller still gains life through combat damage, but I don't take any damage because my Force Bubble's first ability is a replacement effect. And I won't have to sacrifice Force Bubble, since it never will get any counters due to Solemnity.
If I control both of these cards, does that mean I will be able to avoid taking damage? I understand that damage was still being dealt. For example, if creatures attacking me have lifegain, their controller still gains life through combat damage, but I don't take any damage because my Force Bubble's first ability is a replacement effect. And I won't have to sacrifice Force Bubble, since it never will get any counters due to Solemnity.
If you control Force Bubble, you will avoid being dealt damage in any case (the damage won't be dealt to you in any case) (C.R. 614.6). This doesn't change if you also control Solemnity.
As a result, among other things, if a creature with lifelink—
would deal damage only to you, the damage isn't dealt instead (C.R. 614.6), and the creature's controller won't gain life (C.R. 120.3f). You also put that many depletion counters on Force Bubble (or if Solemnity is on the battlefield, you put no depletion counters on Force Bubble, so that it won't have to be sacrificed [C.R. 101.2, 101.3]).
would deal damage to you and another creature, the damage that would be dealt to you is not dealt to you instead (but the damage that would be dealt to the other creature is still dealt) (C.R. 614.6), and the creature's controller will gain only as much life as damage the creature actually deals (C.R. 120.3f). You also put as many depletion counters on Force Bubble as damage that would otherwise be dealt to you (or if Solemnity is on the battlefield, you put no depletion counters on Force Bubble, so that it won't have to be sacrificed [C.R. 101.2, 101.3]).
Recall that if a replacement effect replaces an event (such as "you" being dealt damage in the case of Force Bubble), that event "never happens" (e.g., the damage is not dealt). "A modified event occurs instead[.] Note that the modified event may contain instructions that can't be carried out" (e.g., putting depletion counters on Force Bubble while Solemnity is on the battlefield [C.R. 101.2]), "in which case the impossible instruction is simply ignored" (C.R. 614.6; see also C.R. 101.3).
Does this interaction allow me never to take damage?
Or, does Force Bubble's replacement effect have to happen to avoid me taking damage? (Making Solemnity not useful in this scenario)
In general, if you control Force Bubble, Solemnity is on the battlefield, and you would be dealt damage, then nothing would happen instead — since you can't put any counters on any "artifacts, creatures, enchantments, or lands", including Force Bubble, you don't put any (C.R. 101.2, 101.3, 614.6).
Note that the only thing in this scenario that affects whether damage is dealt is Force Bubble's first ability. Solemnity neither prevents damage nor makes damage impossible.
EDIT (May 15): Clarification.
If I control both of these cards, does that mean I will be able to avoid taking damage? I understand that damage was still being dealt. For example, if creatures attacking me have lifegain, their controller still gains life through combat damage, but I don't take any damage because my Force Bubble's first ability is a replacement effect. And I won't have to sacrifice Force Bubble, since it never will get any counters due to Solemnity.
As a result, among other things, if a creature with lifelink—
Recall that if a replacement effect replaces an event (such as "you" being dealt damage in the case of Force Bubble), that event "never happens" (e.g., the damage is not dealt). "A modified event occurs instead[.] Note that the modified event may contain instructions that can't be carried out" (e.g., putting depletion counters on Force Bubble while Solemnity is on the battlefield [C.R. 101.2]), "in which case the impossible instruction is simply ignored" (C.R. 614.6; see also C.R. 101.3).
EDIT (May 15): Edited.