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If my rogue were to use as a weapon a rapier made of gold, how would the weapon perform?
To answer your specific questions:
Gold is much, much softer than steel. Gold is around 2.5 on the Mohs scale of hardness, whereas hardened steel is 7+. It would rapidly lose its edge and would be very difficult to maintain, and would
Maintaining would definitely be more expensive - a golden sword would bend or break much easier, and would require far more repairs.
A new proficiency would most likely not be required. You can wield an adamantium, mithral, silvered, or other version of weapons without a new proficiency already. Likewise, there's no difference in the rules between, say, a bow made of horn or wood. Proficiency in DnD 5e seems to be based on the broad category of weapon, rather than specific material compositions.
It would be much more viable to coat the blade with gold. Gold and silver have the same hardness, and are very similar in many properties. There are already rules in the game for silvering a weapon, so we can assume the necessary technology to coat a weapon already exists.
It would be a very reasonable house rule to allow you to "gold" a weapon for a higher cost than silvering. The cost of silvering a weapon would represent both the cost of the silver and the cost of the labor/expertise. The labor and expertise would probably be the same cost, but the materials would cost 10x as much (Gold is worth 10x as much as silver by weight). Personally, I'd assume the cost is half materials and half labor, and charge around 5x as much to coat a weapon in gold. The cost should be no more than 10x the cost of silvering
With certain rare exceptions such as adamantine equipment, there is no concept of weapons being made of different materials in DnD 5e. Since the weapons described in the rules don't account for material, you need to make sure with your GM whether the gold rapier is allowed and what properties it follows.
If your GM likes realism in the slightest, they'll probably let you know a gold rapier is a very bad choice of weapon. Gold is an extremely soft metal compared to iron historically used so the rapier would lose its sharp point and warp in normal use very quickly. Its higher mass would be beneficial for bludgeoning weapons, but of negligible advantage for a piercing one.
Personally, I would allow a gold-decorated rapier as a "reflavour" of a normal rapier without any mechanical changes. Changing the effectiveness or proficiencies with the weapon would have a non-trivial effect on game balance if your character relies on the rapier, which is a good reason to not alter it mechanically.
Gold is soft and malleable. One test to see if a coin is really gold is to bite it. Thus, gold would be a very poor weapon as it would bend and could not take a cutting edge. Any alloy durable enough to be a sword blade would not appear to be golden.
A viable alternative would be gold inlay where the blade is mostly steel, but has a design carved into it which is filled with gold.
As gold is highly malleable and ductile it would make a very poor choice for a weapon, it would not retain a edge or a point.
Gold is so soft it can be beaten it very thin sheets to use in gilding things, you could gild a steel or iron weapon people have done in the past. But this is mostly for ceremonial purposes or worn by Kings and other important figures that would not actually fight an enemy. The gold would be quickly worn away in actual use and need to be reapplied.
tl;dr- While pure gold's too fragile to be a good material for swords in general, it'd be especially problematic in rapiers as they're already prone to breaking when composed of stronger metals.
Rapiers are already light, thin blades:
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They're also prone to breaking:
How strong were rapier blades, couldn’t they break easily?
There is substantial historical evidence for rapiers breaking during fights (in bodies and against other weapons), and I have held several real ones that had broken or even bent points. I’ve examined many others that to me felt so thin and light I am sure they would readily break if used to slash with or even if seized by a hand and forced to bend.
As for gold as a working material, pure gold's doesn't tend to resist deformation very well. It's easily enough deformed that even fine jewerly uses an alloy of gold with other metals instead.
So, while a golden rapier might make for an interesting decoration, it wouldn't seem to be a reliable weapon.
Rapiers already aren't general-purpose weapons; instead:
What kind of sword is a rapier?
The best answer I can give is that the true rapier is a long, narrow, rigid, nearly edgeless single-hand thrusting blade with a thick, tapering cross-section and very narrow and sharp point. There is no question rapiers vary in their shape, length, and width and especially in their hilt configuration. But rapiers are generally thin, light, fast, and well-balanced thrusting swords intended for unarmored single-combat.
This is, it's already a bit of a stretch to use a rapier in D&D-style combat since it's really just for "unarmored single-combat" between humanoids; rapiers weren't meant for stuff like attacking armored opponents, non-humanoids like dragons, etc..
Since D&D's already stretching things by having rapiers being used as general-purpose weapons, it's probably best to avoid stretching it further — unless your group has no concern for this sort of thing, in which case, I suppose you could house-rule that gold has properties entirely distinct from those found in real-life or something.
