1) Directly, no resistor required for either the A2982 or ULN2803A.
2) Both the A2982 and ULN2803A can inherently handle reverse voltage, so D1 isn't needed. But you've misinterpreted the function of the internal diode, as it's not what's responsible for blocking reverse voltage. Look closer and you'll see it's connected between the output and ground - same as your D2, and in the same orientation, on both ICs. It's meant to handle inductive kickbacks from relay coils. So technically, D2 isn't strictly necessary either. Though if you're trying to minimize RF noise, or there's a wire longer than a few inches between the IC and the relay, I'd include D2 close to the relay to shorten the path that the inductive kickback has to travel.
3) If the datasheet says to tie unused inputs to ground, and add 10K pull-down resistors to used inputs that may not be actively driven (like during MCU startup), then just do it.
Even if you find it works without the pull-down, which it probably will under the most circumstances; but shortcuts like these can come back and bite you when you least expect it.
4) Both the A2982 and ULN2803A are suitable.