I know that, as a cloaked spy, I can recharge my watch from a dispenser. In fact, a L3 dispenser can recharge my watch faster than it drains. However, I know that dispensers display a glowing tether to units that they are resupplying. If I cloak next to (or on top of) an enemy dispenser, will I get tethered and thus be revealed to detail-oriented Engies and Pyros?
No, cloaked spies will not be tethered by dispensers.
It's the same mechanic as to whether medics can attach medigun beams to cloaked spies. They can't, and I know flickering from being bumped doesn't change it.
Most special effects are hidden while cloaked, with notable exceptions being milk/jarate/blood/water drips and fire.