[RPG] Do acid and sonic damage bypass object hardness

RAW, it probably does 0 damage, because nothing says otherwise

It says “half,” so you probably round both sides down and get 0. Note that the “minimum 1” rule is for a hit specifically, and this spell doesn’t involve one.

That said, every group I’ve played with has allowed the caster to choose which way the odd damage went in a half-and-half situation. Usually, it’s 1 damage, who cares? And certainly the minimum 1 thing should also apply to non-hit damage, as here. With thunderhead, it is the entire spell, though, so I wouldn’t allow it to apply to each half, which would effectively double thunderhead’s damage. Alternating bolts of electricity and sonic damage probably makes the most sense.

The stunning effect comes when the spell “concludes”

I actually don’t think it’s either the first or each bolt, but rather the last bolt, when the spell is complete.

That said, I don’t think the term “concludes” is actually defined, and it’s conceivable that they meant the casting of the spell is complete, so it would happen as the first step. Either way, though, I don’t see any way to justify it on each bolt.

No, it doesn't bypass hardness at all.

(This may have been different in D&D 3.x, but it isn't the case now.)

The acid descriptor rules don't suggest acid damage gets any special handling, nor do the rules on hardness. There's rules on what energy damage does to objects, though - and from the notes on damage in d20pfsrd, acid damage is considered energy damage - but these rules don't add anything about bypassing hardness:

Energy attacks: Energy attacks deal half damage to most objects. Divide the damage by 2 before applying the object's hardness. Some energy types might be particularly effective against certain objects, subject to GM discretion. For example, fire might do full damage against parchment, cloth, and other objects that burn easily. Sonic might do full damage against glass and crystal objects.

In fact, this suggests energy (including acid) damage deals less damage to objects usually: you halve the damage before doing anything, and half damage is the most it can deal, and then you apply hardness, so it might deal even less than that.

At GM discretion, energy damage can still retain its full damage amount depending on the situation, such as if you're using fire on wood. It's ambiguous whether they mean "deal full automatic damage, ignore hardness" or "just don't halve the damage before applying hardness", but since this is an explicit written author's suggestion to ignore the rules, trying to figure out to what extent the rules are advising us to ignore them is probably not worthwhile. Just use your discretion and break them whichever way you prefer given the situation.

Other than GM discretion, there's no damage bypass, nor half-damage bypass, etc.