Sure, nowadays many "jazzers" also play with distorted sounds (they tend to lean more towards the Blues Setup realm), but here, we're mainly focusing on the traditional clean sound department.
Absolutely - semi-acoustic guitars or even fully resonant pieces with humbuckers remain the standard here. A clean sound just resonates much more vividly and feels more "woody" when it's genuinely brought to life by air.
For the traditional jazz sound, where no distortion is sought after, transistor amplifiers have long been established, particularly the legendary amps from Polytone. The key is having a very responsive playback and a smooth, warm tone. Jazz purists have a somewhat peculiar attitude toward their equipment (excluding the guitar): as long as it sounds good, we'll handle the rest with our fingers. This also applies to...
Less is more here. The chorus effect has now become the standard, often in stereo with two amplifiers, for instance. A compressor can give the solo tone a bit more punch, but too much might take away the desired dynamics. Most are content with reverb from the amp.
The jazzer needs only one sound: their own! So, neck pickup engaged, tone knob slightly rolled back, subtle chorus, and away you go.