This is possible, but you would then need PC software to process the data stream. Some speed constraints to consider are as follows:
USB on the Nano conforms to USB 2.0 full-speed (12 Mbit/s). SD write speed is roughly the same. Sustained end-to-end speed for either interface (USB or SD) however is likely to be significantly less. Maximum sampling rate on the Nano is currently (V3.40) 16 Mbit/s.
The functionality of the signal generator is the same as for V3.3x which was confirmed to work also for V2 hardware. Hardware version is displayed on the startup screen (V1 or V2) so you can verify that your Nano is identified correctly.
Personally I think the double lines obscure the displayed waveform and so prefer the single level line (sensitivity is more of a set-and-forget value). This of-course is a subjective preference and so an option to choose may be the way to go.
The PULS measurement is a count of low-to-high (below/above trigger) transitions across the full capture buffer.
Any single capture will use the full buffer capacity (currently 4098 points) with the trigger position shifting left/right within 1/3 of the buffer between individual samples. In order to keep the trigger centered on screen and still allow every single point to be viewed there will be empty space (no data) far left and/or far right on the display. This empty area is outside of the buffer and size is related to trigger position.