Arcane duelist lets me use the hand holding my bonded weapon for semantic components of spells. Does this mean I can use it to make touch attacks? (Not in addition to a regular attack, but just touching them with the weapon)
No, it never say this.
If not, does this mean that I have to have my other hand free to make a touch attack, or can I make a touch attack with any part of my body?
You typically need a free hand.
Does a touch attack have to be made with bare skin, or can I make one while wearing gloves?
You can do it with gloves, even with spiked gauntlet.
Vanish lasts for 1 round/level (up to 5 rounds). If I use the Extend Spell metamagic feat / rod to double the duration, does this apply before or after the duration is capped at 5?
After.
Making a weapon out of cold iron cost twice as much as normal. Does the price double before or after you add the +300gp to the cost for making it masterwork?
After, iirc.
Does playing a merfolk have an unwritten consequence of losing your feet magic item slot since you don’t actually have feet to wear footwear on?
She don't have feet, so she get all the obvious consequences.
I believe it's derived from this FAQ.
It stemmed from this discussion.
To quote the final post from that thread (not a developer, but in summation of the thread as well as the FAQ):
Given that wearing a gauntlet-like weapon will not accidentally discharge a held touch spell
AND
Given that touch spells may be made by a hand wearing a gauntlet
AND
Given that touching someone with the hand holding the charge will discharge that spell whether you like it or not
THEN
Making a normal weapon attack with a spiked gauntlet that is currently holding the charge will also discharge the held touch spell (if the attack hits)
Arcane duelist lets me use the hand holding my bonded weapon for semantic components of spells. Does this mean I can use it to make touch attacks? (Not in addition to a regular attack, but just touching them with the weapon)
No, somantic components are not touch attacks.
If not, does this mean that I have to have my other hand free to make a touch attack, or can I make a touch attack with any part of my body?
You need a free hand.
Does a touch attack have to be made with bare skin, or can I make one while wearing gloves?
Yes.
Vanish lasts for 1 round/level (up to 5 rounds). If I use the Extend Spell metamagic feat / rod to double the duration, does this apply before or after the duration is capped at 5?
After. Extended vanish with a 5th level caster lasts 10 rounds.
Making a weapon out of cold iron cost twice as much as normal. Does the price double before or after you add the +300gp to the cost for making it masterwork?
Before.