You think video output does not work via such an USB-A to -C Gen 2 cable or adapter?
(like this one
It definitely won't, as alt modes (such as DP alt mode) are specific to USB-C and there is no way of carrying this signal through a type-A connector. The required wires simply aren't there.
For type-C cables, there are two sets each of RX and TX differential pairs in the connector/cable for high speed signalling, which for G1 modes are fed through a multiplexer depending on the direction the cable is inserted so that the signal works regardless of how the plug is inserted (i.e. only one pair of each is used at a time). Nothing changes for G2 except the signalling rate. For G2x2, all pairs are used simultaneously, effectively doubling the lane count. This means USB 3.2G2x2 (ugh) is incompatible with type-A connectors.
The changes in the signalling are also the reason for G2 cables being shorter, as the more complex signal degrades faster and is more susceptible to interference
Edit: To clarify, (DP and similar) alt mode signals are sent through the TX/RX high speed data lanes in the type-C cable that are left unused by G1/G2 muxing. As these lanes don't exist on the type-A connector it is entirely unable to transmit these signals. This also means that 3.2G2x2 is incompatible with alt mode, meaning one has to pick either double lane count for higher data speeds or alt mode for other things.
Also I have another question, if both my mobo and device is Gen 2 (10Gbps), and I use Gen 1 (5Gbps) to connect both, will I get the full speed of Gen 2? Been Googling these few days and can't find the definite answer because there is no comparison of pinouts of the 2 gen, and also not sure if wiring/internal structure of these 2 are same or not
There is no change in the pinout between G1 and G2, only changes in the signalling. 8-pin Type A 3.0/3.1G1/3.2G2 has the required TX and RX pairs to transmit this same signal.
Edit: As such this means the transfer speed comes down to the cable's ability to maintain signal integrity. For G2, that limit is rather short due to the complexity of the signal. Wiring is otherwise identical.
Edit(s): see above.