I'm a kayaker but I have gone out with SUPers. This is what I see:
How does it work if e.g. you paddle down a river so don't go back to where you started?
Most people come back to where they started, against current & wind if they have to. If you want to go 1-way, you have go organise transport back or getting dropped off.
What type of pump do you use?
Most people operate pumps on their car battery (cigarette lighter port?)
Do you put the pump and carry bag inside a waterproof bag and lash to the board?
They could do. You could have someone to drop you off. Before they drive off, Inflate the pump on their car battery then they meet you at the end point. Use the pump to deflate too, possibly. They could carry your trolley if you need one (not sure why you'd need one, everyone has inflatable boards & they weigh nada). If it's hand pump ( I just sold one!) it can handle getting wet.
SUPing is hugely sociable in my observation, you should be able to on Facebook find a local social group that goes out a lot.