Welcome!! I'm sorry to hear about your mother. Coins lose their numismatic value after they are put into jewelry unless they are very rare. They are worth spot gold price.
Obviously your Gold American Eagle, which is basically a bullion item anyway, is worth about $488. The Austrian piece is most likely a 3.4909g .9860 fine gold ducat (KM#2267), a trade coin. I can't make out the date in your picture, but they were minted from 1901-1914, and there were also restrikes in 1915 and 1920-36 bearing earlier dates. I don't know the diagnostics to distinguish the real from the restrike, but since it's a jewelry item, the bullion value of about $53 applies anyhow. For undamaged pieces, the values are in the $60-200 range, depending on grade. If the coin is much larger than it appears in the photo - about 40mm - it's a 4 ducat, KM#276, so just multiply all the numbers by 4.
And don't forget that the bezels are likely gold too. You're best bet may be to sell these at a jewelry store. Since they are already set in bezels - a jeweler might even give you a bit more for them
