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All motherboards work with all PCIe GPU's. PCIe is a platform agnostic standard. If your motherboard is old enough that it does not support PCIe 4.0 though, you should definitely upgrade, because that probably puts your CPU as being old enough to bottleneck a 4070.
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Again, platform agnostic. there are some internal benefits for running all-AMD hardware, but they're minimal. Intel will work perfectly well. Refer to point 1 to see if you should upgrade or not.
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Generally they mirror Nvidia's numbering scheme. Aka, the x900 GPU's are the halo card, followed by x800, x700, x600.
Unfortunately AMD's current lineup is a little behind in the midrange market. Their only midrange cards in the current lineup are last-generation RDNA2 6000 series cards aside the 4060-targeted RX 7600xt, which puts the closest AMD equivalent to the 4070 to be the RX 6900xt/6950xt. The 6950xt is currently floating around that 600-700$ pricepoint that the 4070 is at, and roughly matches it in raster performance.
Raytracing is significantly worse though, it uses more power, and AMD's software stack (frame interpolation, recording/encode, etc), while usable, just isn't quite up to the same par as Nvidia's. So if any of that really matters to you, I unfortunately have to recommend the 4070.
However, if you can handle those downsides and just need to play traditional raster games, please please buy the AMD card and do not feed Nvidia's idiotic greedy monopoly machine. Jensen is smoking crack and the last thing anyone needs to do is feed that man's overinflated head more fucking cash.
Or even better- wait until 2024 if you are not in need of an immediate upgrade. Intel's first generation Arc GPU's had a rough launch but have improved massively over the past year with driver upgrades, to the point of being fantastic first generation products. Their second generation Battlemage GPU's are expected at the beginning of 2H 2024 and could potentially be a killer value in the midrange segment.