Sorry for making this so complicated...
When buying a PC with older or used parts to build yourself, you need to compare sets of mainboard+cpu+ram. Like Nintendo Maniac 64 said several times, there are mainboards that support certain cpus, but only on certain firmware versions. Since most mainboards, especially cheaper and older ones do not come with usb flash support(firmware udating without a cpu installed), you need to be careful of that. And then you need to get RAM that fits into that mainboard, so mostly either DDR3 or DDR4. All other components should be switchable.
Intel cpus since haswell, so the G3258 or the i5-4670K, i5-4690K, i7-4770K and i5-4790K should be fine for Dolphin, preferably an i3, i5 or i7. Cpus with Pentium in the name might be ok, but Celeron and Atom in the name should be avoided, as they have cut down features that can make a day and night difference for Dolphin. As for amd cpus, super avoid everything before zen. Nintendo Maniac 64 said that zen1 is also bad for Dolphin, i don't know myself, but i have no reason to doubt that. So for an amd cpu, you want zen2 or later. With amd cpus, you need to consider that most of them don't have integrated graphics, but it seems you already knew that.
What i mean is, it's possible you might find mainboard+cpu+ram for an i7-6700 for less money than a i5-4670(K), and then you should definitively go for the i7-6700. And some Xeon cpus might be much cheaper than they should be and are really good deals.
Last but not least, you might find a good deal on a complete office PC that's cheaper than what mainboard+cpu+ram cost separately. Pricing on the used market is strange sometimes. With those PCs you need to be careful if they use standard parts though. Sometimes the power supply and mainboad are non standard and only work together or there are other weird things. That's fine for the most part, but might be a problem if you want to upgrade in the future and get a proper GPU.
When buying a PC with older or used parts to build yourself, you need to compare sets of mainboard+cpu+ram. Like Nintendo Maniac 64 said several times, there are mainboards that support certain cpus, but only on certain firmware versions. Since most mainboards, especially cheaper and older ones do not come with usb flash support(firmware udating without a cpu installed), you need to be careful of that. And then you need to get RAM that fits into that mainboard, so mostly either DDR3 or DDR4. All other components should be switchable.
Intel cpus since haswell, so the G3258 or the i5-4670K, i5-4690K, i7-4770K and i5-4790K should be fine for Dolphin, preferably an i3, i5 or i7. Cpus with Pentium in the name might be ok, but Celeron and Atom in the name should be avoided, as they have cut down features that can make a day and night difference for Dolphin. As for amd cpus, super avoid everything before zen. Nintendo Maniac 64 said that zen1 is also bad for Dolphin, i don't know myself, but i have no reason to doubt that. So for an amd cpu, you want zen2 or later. With amd cpus, you need to consider that most of them don't have integrated graphics, but it seems you already knew that.
What i mean is, it's possible you might find mainboard+cpu+ram for an i7-6700 for less money than a i5-4670(K), and then you should definitively go for the i7-6700. And some Xeon cpus might be much cheaper than they should be and are really good deals.
Last but not least, you might find a good deal on a complete office PC that's cheaper than what mainboard+cpu+ram cost separately. Pricing on the used market is strange sometimes. With those PCs you need to be careful if they use standard parts though. Sometimes the power supply and mainboad are non standard and only work together or there are other weird things. That's fine for the most part, but might be a problem if you want to upgrade in the future and get a proper GPU.