Electrical – How to install an electrical panel in the shed

Since the garage panel is a subpanel and not a service entrance, you should be able to turn the entire subpanel's feeder off upstream -- I'd lock it out with a breaker-lock + padlock myself to avoid a shocking surprise from somebody else in the family.

With that done, running the gazebo circuit into the subpanel should not be an issue -- it's like running wires into a giant junction box in that you'll need to support and clamp them in the same way as you would a box. Once you have the wires in there with enough slack to make the connection -- simply screw the bare wire to the ground bar, the white wire to the neutral bar, and the black wire to the screw terminal on a spare 20A breaker, presuming you have one. You'll also have to take the front cover (dead front) off and put it back on, but that shouldn't be too hard.

Once everything is hooked up, put back together, and ready to go, simply take the lock off the feeder breaker and turn it back on.

Go ahead, but make sure to get the details right

Since you are pulling 25A (out of the 40A total available) on the pool feeder right now, you can route a single 20A/120V circuit (using 12/2 W/G UF or 12AWG THWN hot/neutral/ground in the conduit) from the pool subpanel's spare breaker to a disconnect at the shed (any standard A/C disconnect box will do the job happily), then you can wire the shed lights and receptacles to the disconnect box as they won't consume but a fraction of the remaining 15A normally.

You won't need a full subpanel at the shed since you're feeding it with a single branch circuit, nor will you need a grounding electrode as you are installing a disconnect for a single circuit, not a feeder and subpanel. (This is per the exception to 250.32(A).)