You can’t really force your neighbor to put up a fence unless their fence is on violations of some code (for example most municipalities have rules about fences around pools and such). Even then the best you could do is report it to whatever force handles building code violations and then wait to see what they do (they may force the fence to be brought up to code or they could just charge a fine and allow it to stay as is).
What you can do, short of building a second fence, is put up a privacy screen or with permission modify it in some other way.
As for the jumping, you can take photos or videos and report it each time it happens. If you need to leave your dog outside unattended for whatever reason, a dog run may be a cheaper option than building a new fence entirely.
May want to try privacy fence like this: https://www.amazon.com/dp/B001OJXVKW/ref=cm_sw_r_cp_apa_glt_fabc_AFHM6MTWPM4RDANWH0XG
You can attach it to the fence or remove it when needed.
What about that construction fence style fabric? You could ziptie it on the grommets.