Does Hold Person cast on a Spider Climbing NPC affect their ability stay on a vertical surface?

Question

An NPC spell caster that is using spider climb at the top of a 30' pillar where it meets the ceiling is magically held by hold person. I have two related questions:

1. When Hold Person goes into effect, does the NPC fall, or does the Spider Climb magic (which is still in effect) keep his hands and feet bound to the pillar and leaving him suspended 30' off the floor?

2. Assuming he does not fall immediately, if the party shoots him with an arrow or other ranged attack, since Dex checks auto-fail when Paralyzed, would that knock him off (and thus fall for 3d6 damage).

Answer 1

Hold person paralyses the NPC, so they fall

According to the situation is:

An NPC spell caster that is using spider climb

This indicates that the NPC has cast the spider climb spell on themselves.

However, the NPC automatically loses concentration on spider climb if paralyzed by hold person due to being incapacitated (Basic Rules, p. 97):

The target must succeed on a Wisdom saving throw or be paralyzed for the duration.

The description of the Paralyzed condition says (Conditions, Appendix A):

• A paralyzed creature is incapacitated (see the condition) and can’t move or speak.

That by itself breaks concentration (Basic Rules, p. 84):

You lose concentration on a spell if you are incapacitated or if you die.

Failing the saving throw against the hold person spell causes the NPC to fall unless something else is keeping them up there.

There is no Dexterity check involved

The issue at hand is the NPC's concentration on the spider climb spell ending. If they successfully save against hold person, the PCs may still knock them down with the arrows' damage.

Spider climb has a duration of "Concentration, up to 1 hour", and its description states:

Until the spell ends, one willing creature you touch gains the ability to move up, down, and across vertical surfaces and upside down along ceilings, while leaving its hands free. The target also gains a climbing speed equal to its walking speed.

If the PCs hit the NPC after a successful save against hold person, the damage will induce a Constitution saving throw to maintain concentration (Basic Rules, p. 84):

Whenever you take damage while you are concentrating on a spell, you must make a Constitution saving throw to maintain your concentration. The DC equals 10 or half the damage you take, whichever number is higher. If you take damage from multiple sources, such as an arrow and a dragon’s breath, you make a separate saving throw for each source of damage.

If the NPC fails that saving throw, then the NPC will fall. If they pass, they'll stay up there.

Two arrows hitting is better than one.

Answer 2

No, unless losing concentration terminates a critical spell.

Nothing about being paralyzed would inherently cause a spider-climbing character to fall; they'd be paralyzed in place on whatever surface they were clinging to. Paralysis causes you to be unable to move, but does not directly alter any effects that allow you to cling to surfaces.

The paralyzed condition does cause the target to become incapacitated, which automatically ends concentration on any spells they might be holding, so if you paralyzed a caster who was using the spider climb spell on themselves, they would fall because the spell ended. But if the climbing character got a spider-climb-like effect from some other source, such as from a magic item (say, slippers of spider climbing or a cloak of arachnida) or from a separate caster, then a loss of concentration would not have any particular effect.

You specifically said a spellcaster is using the spider climb spell, so in this case they would fall, but that's because of the way paralysis interacts with concentration, not due to a direct effect on the ability to cling to surfaces. You could just as well say "paralysis destroys barkskin" -- technically, yes, it does break concentration and end the spell, but it isn't directly affecting barkskin or spider climb in specific.