Looking at that Chinese datasheet it seems the chip itself would be capable of “isolating” the logic side from the driver side but if you are using a premade board as shown in your second image, then this won’t be possible and your GND rails for the controler and solenoid have to be joined (as they are already on the board).
I also don’t see sufficient kickback protection, so you better add a flyback diode (anti-parallel to your solenoid) - with a bi-directional motor you’d need something like this the middle part of this diagram (the the MX1508 takes the place of all the BJTs)
(coutesy of www.circuitstoday.com)
ad3) yup (although an H-Bridge is somewhat overkill for a mere solenoid - a humble BJT or FET would be enough).