[RPG] How much do multiple weapon enchantments cost

With base price being defined as how much the item is priced in the store, and not the discounted cost for crafting the item.

Correct.

Items you can buy from the store usually have a CL listed.

For example, the Ring of Invisibility has a CL 3rd, so a player would need to take a DC 8 spellcaster check to successfully craft the item.

No, oddly enough. The crafter of the item sets its caster level, from a minimum of whatever it takes to cast the requisite spells (or other requirement listed for the item), to a maximum of the crafter’s own caster level. Since caster level typically costs money, increases DCs, and so on, most crafters use the lowest caster level possible for the item.

The caster level listed with items is the “typical” caster level for that item, where “typical” is more-or-less just something the authors made up. For most items, it is the minimum (e.g. that ring of invisibility, requiring as it does the 2nd-level invisibility spell, which has a minimum caster level of 3rd), but there are exceptions (e.g. sovereign glue, which has an absurd listed CL of 20th, despite only really requiring 3rd for make whole).

From what I can gather, the cost to craft a magical item with multiple abilities costs the full price for the most expensive bonus, then 1.5 times the price of each additional bonus.

Correct.

It is worth noting that D&D 3.5, upon which Pathfinder is based, added a rule in Magic Item Compendium that certain, basic sorts of bonuses do not incur this premium. For instance, making your ring of invisibility also include a deflection bonus to AC (à la ring of protection) would not cost extra (just the cost of ring of invisibility plus the cost of ring of protection), because deflection bonuses to AC were one of the “generic” bonuses you could have on rings. Other examples included enhancement bonuses to ability scores, resistance bonuses to saving throws, etc.

This change allowed for characters to get their critical math fixes, while still allowing them to get “fun” and interesting items. It led to a much smoother game that penalized characters less for being responsible and buying the critical, but boring, +number items.

I will admit that Paizo not only has not ported this rule, but adamantly opposes it with its recommendations. Paizo considers it important that characters pay extra for combining such items. I will state flat-out that they are quite simply wrong. This attitude massively, and unnecessarily, shafts the classes that were already weakest. I cannot more strongly recommend that you ignore them on this issue.

Lets say I want to craft a Ring of Invisibility and also enchant it with Magic Aura so that it registers as a non-magical ring.

Your example is done correctly.

1. Does adding the Magic Aura effect to the Ring of Invisibility increase the final Caster Level of the ring, and thus the DC spellcaster check? If so, by how much? Just 1 since I used Caster Level 1 to add the effect, making the ring a CL 4th with a DC 9 spellcaster check?

Caster level requirements are minimums, so use the highest minimum as the overall minimum of the item. In this case, magic aura requires CL 1st and invisibility requires CL 3rd, so the ring requires CL 3rd. You could craft with a higher CL (requiring a higher DC), which would make the ring more resistant to dispel magic et al.

2. When you upgrade a magical item or add additional abilities to an existing magical item, do you take a DC spellcaster check at the end of the crafting time? The rules do not state this outright and I've not been able to find an answer.

   1. If you do, do you use the CL of the new ability for the check? With the example for adding Invisibility to a ring of protection, would the DC of the spellcaster check be that of the CL of the ring of Invisibility (3rd), or something else?

The DC would be based on the item’s CL, whatever it is. At a minimum for this ring, 3rd.

Does the CL of the item increase when upgrading an item? Bracers of Armor has a CL 7th regardless of the strength of the enchantment bonus. Would upgrading the bonus from +1 to +3, or +1 to +5, still use a DC 12 spellcaster check?

If you were correct about bracers of armor requiring CL 7th regardless of enhancement bonus, you would be correct. I believe there may be some examples where this would be the case.

However, the bracers of armor do not require CL 7th. Rather, they require that

creator’s caster level must be at least two times that of the bonus placed in the bracers, plus any requirements of the armor special abilities

There is evidence it needs to be magical.

According to Page 6 of the Magic Item Compendium (the book where a lot of those +gp cost special abilities are):

To add a special property to a shield or suit of armor, the shield or armor must already have at least a +1 enhancement bonus.

This is pretty cut and dry that every single special ability, from Acidic (the first one), that has Price: +1, to Agility (the second one), that has Price: +500 gp, requires it to be magical.

I would require it to be magical. Just to be consistent to what I believe the intent is. Most of those special abilities perhaps don't warrant a +1 through +5 cost, but nonetheless require the item to be "special" before it can become "more special."