Espeon

Can Espeon's "Solar Suggestion" attack place damage counters onto a Jolteon that can't be hit by effects due to a successful coin flip during its "Agility" attack, or onto a Toxicroak [G] with its "Anticipation" Poke-BODY enabled? If I can't place them, what do I do with the damage counters I removed from my Pokemon?

You cannot place damage counters onto that Jolteon or Toxicroak [G], but you can still remove damage counters from your own Pokemon. If you select a Pokemon under the status of not receiving effects, the damage counters that would be placed onto that Pokemon simply disappear from play.

If I choose 4 damage counters on my Pokemon with Espeon's "Solar Suggestion" attack, can I then place 3 or fewer damage counters on my opponent's Pokemon?

No, you can't. You must place the same number of damage counters on your opponent's Pokemon as you removed from your own Pokemon.

When using Espeon's "Solar Suggestion" attack, do I have to move all four damage counters from one of my Pokemon to one of my opponent's Pokemon, or can I split the damage counters from multiple different Pokemon on each player's side?

You can choose up to 4 counters from as many of your own Pokemon as you like, and can split them up on your opponent's Pokemon in any way you want.