Could identifiable satellites be put in orbit that could help someone on the ground determine longitude without telescopes or too much calculation trouble? Would you always need a timepiece, or might this be done only geometrically?
Yes, you could use such a thing to calculate your longitude geometrically, from any place where the satellite is visible (so, approximately half the world).
The method is identical to that used to calculate latitude by sighting the sun or stars. The trouble is that the sun and stars appear to move east to west, so you don't know the east-west component of their absolute position unless you know the time.
A geostationary object has a different problem: you always know its geocentric position, because it's always the same, no clock required... but a single sighting can only narrow down your position to somewhere along a circle on the Earth's surface. If you also know your latitude, that narrows your position down to 2 possible points, which will usually be very far away from each other, so it shouldn't be difficult to figure out which one is correct, and thus what your longitude is. If you can see two satellites, two sightings will give you your longitude exactly, no guessing required.
The more satellites you have, though, the harder it will get, because determining your position requires being able to accurately identify which satellite you are sighting. The ideal situation would be to have exactly two satellites visible high in the sky from anywhere in the world. You can't manage that perfectly, but a ring of 5 satellites would work pretty well. Perhaps you could fit them with high-power spotlights (illuminating only the hemisphere from which they are visible, rather than wasting power illuminating all of space) of 5 distinct colors--or just 3 distinct colors, so that any pair of simultaneously-visible satellites is distinguishable, as long as you can can determine which way is north vs south.
Geostationary orbits around Earth are perturbed by lunar gravity, solar gravity, and the obliquity of the Earth, and so require approximately 50m/s of station-keeping delta-V per year.
