The original explanation focused on the vortex production. For a description of what's happening when the vortex breaks off into child vortices, I've added this new section.
The breakup of a single vortex ring into many smaller rings looks sounds like it should be driven by an azimuthal instability. Indeed, under certain conditions, azimuthal waves can grow around a ring's circumference, with instabilities showing as many as 20 peaks around a circumference [1A].
With this in mind, consider what happens when these instabilities drive two vortex filaments closer together. In an inviscid fluid, the filaments could pass through each other and continue developing. But in a viscous fluid, diffusion can cause the overlapping filaments to reconnect into different topologies, spawning child vortices. Such behavior is explained in [2A, 3A] for an elliptical vortex ring splitting into two vortices, but the same mechanism could drive the breakup for higher-frequency instabilities.
(Note, the initial instability in the vortex ring would likely be initiated by the drop-splashing dynamics, and would thus be governed by factors like surface tension, impact speed, etc.)
1A. Green, Sheldon. Fluid Vortices. 1995. p124-129
2A. Ibid. p134-140
3A. Dhanank and Bernadinis. J. Fluid Mech., 109. p189-216. (1981) DOI: 10.1017/S0022112081001006
Cool observation! Seems like it would make an interesting research project. It's hard to explain exactly what you saw without pics/video, but here are some initial ideas/background on the problem based on some research in the area of droplet splashing:
Why does this happen?
A lot of researchers dating back to the 1800s have observed regimes of droplet impact can generate subsurface vortex rings (see all refs. below). The driving mechanism is a balance of drop inertia, water viscosity, and surface tension. Somehow, these dominant mechanisms should be able to explain the vortex ring dynamics you see.
Morton et al.  use computer simulations to show that droplet impact on a surface can generate multiple vortex rings under certain conditions. Furthermore, if a small jet or other drops from the lemon impacted the water after the initial drop does, you could have additional vortices formed.
- Interaction of any of these multiple vortices could lead to complex break-up events.
A lot of the surface tension effects have been studied by many researchers including  and  below. The branching of the vortex rings sounds reminiscent of common surface-tension driven instabilities. Going off of some of the above research, one hypothesis could be that the combination of surface waves and collapsing unstable jets drives vortex breakup to follow the patterns you see.
How deep would the vortex rings go?
This should depend on both the viscosity of the water, the speed of impact, and the surface tension of the water.
An earlier study  (confirmed by others later, eg ) demonstrates that the vortex penetration distance is affected by the droplet shape when it hits the water surface (since the drop oscillates around a spherical shape). This could potentially also set the stage for a directional vortex breakup like you saw.
Given these mechanisms, I would be surprised if the vortices broke up more than once or twice in the cup.
- Morton et al., Phys. Fluids 12, 747 (2000) DOI: 10.1063/1.870332
- Cresswell and Morton, Phys. Fluids 7, 1363 (1995) DOI: 10.1063/1.868524
- Thoraval et al., Phys. Rev. Lett. 108, 264506 (2012) DOI: 10.1103/PhysRevLett.108.264506
- Rodriguez and Mesler (1987) DOI: 10.1016/0021-9797(88)90414-6