1. No, it's not continuous, except at lower sample rates). At 3.2 GS/s and 12 bit samples, that's about 40 Gb/s (after error correction overhead etc.). That would challenge even a USB 4 connection. There is another open source scope, Thunderscope, yet to be released, which does this but at 500 MS/s and 8 bits.
2. The opto-relay is a traditional choice for AC coupling. It is cheaper ($1 vs $10), smaller, has less insertion loss, uses less power, has less capacitance, etc. than a mechanical relay. And it does NOT have to have large bandwidth - the high freq part is always carried by the parallel capacitor. The opto-relay just switches through the low freq part of the signal, down to DC. Good question! Dave should do a video on this.