is this much engine movement normal?

https://www.youtube.com/watch?v=XZx1lTV9No4

yes you can see the engine moving, especially when the gas pedal is applied.

For some time now there's been a roughness, a slight growling/gurgling (exhaust related maybe?) when accelerating. Engine seems to hesitate a bit sometimes when accelerating, particularly when you're coasting for a bit, then reapply gas, and also when you shift from park and just let it roll.

I realize cars weren't made to receive acceleration with the brake is applied, but that seemed a bit excessive to me.

2002 Civic EX, 66,700 miles. Mom's little buggy (she's almost 83 and perhaps a little more lead footed then is good for her!).

It doesn't seem normal. That first change from park to reverse seemed a bit harsh, then that flexing with the brake applied revving the engine seems a bit odd too. A little bit of flex is normal, but that seems like more than normal.

i would say your rear mount/torque strap by firewall is bad or wearing out for the movement. it also flexes your exhaust quite a lot and may leak to a crack in the pipe making it louder. there may also be another center mount by radiator thats worn out, not sure if that car uses 2 or 1 to keep engine from rocking when shifted into gear

It looks a little excessive to me. Luckily, engine mounts or even mount inserts are pretty cheap. The expensive part is getting them installed, it's not a fun job.