1.) My cigarette lighter outlet doesn't stay on after the ignition is
turned off, so my only recourse is to permanently connect the solar
panel to the battery posts.

Likewise, the car I use it on has the cig ligher off when the car is
off. What I do is to connect the alligator clips directly to the
battery terminals. The cord is long enough in my case that I can put the
solar panel inside the windshield.

2.) The alligator clips for the battery posts are too obtrusive and
would interfere with my current battery post connectors. There needs to
be a solar trickle charger with better integrated battery connectors.

You'd probably need to fab something up to fit your requirements.

I'd like a manufacturer to create a rechargeable trickle charger that
could be easily removed for recharging a few hours at home, then
reconnected to the car to last a week or month of trickling. Of course,
the charger brick could be removed without disconnecting the alligator
clips from the battery.

3.) The instructions to these devices state that in order to prevent
damage to the trickle charger, it's supposed to be disconnected before
starting the car and while driving. So how am I supposed to disconnect
it from the battery before starting the car? Others have said that it
works fine while starting the car.

I just pop the hood and disconnect it.

That sounds primitive and inconvenient, but that's exactly what I'm
doing right now with my battery cutoff switch. The problem with using
the battery cutoff switch to prevent overnight drain is that it resets
the clock and the OBD computer at each morning startup.