How many hands free does one need in order to use a Healer's Kit?  How many object interactions and actions (and turns) would it take for a person with an object in each hand to put one or both objects away, take out the Healer's Kit and use it, and then put away the Healer's Kit and draw one or both objects again?

I believe you just need an action to use it, you don't need to really worry about the interactions.

Turn one, you pull out the healers kit and use it. I would say it uses one hand, personal preference.

Turn two, you can put it away and pull out your weapon/shield again to use it.

You know, I never thought of this before, but... Healer kit probably requires 1 free hand (it'd be a little tough to use that way, but this game yada yadas over a lot of that). Sheathing a weapon and retrieving the kit both take an interaction, but you only get one. The general consensus is that you can drop your weapon on the ground without using an interaction, and you can pick it back up using an interaction next turn, but you also have to stow the kit which takes an interaction.

So, you can be back int the fight by turn 2 by:

1. Dropping your weapon (free), retrieving the kit (interaction), using the kit (use item action).
2. Leaving kit on the ground (free), grabbing weapon (interaction), and use your action for whatever (attack for example).
3. Pick up your kit after the battle (a cruel DM may have smarter enemies try to break the kit during battle, or may have enemies try to steal your weapon in the turn you are not holding it).

Playing it safe would take longer, a full 2 turns without a weapon and not stabilizing your ally till the second turn:

1. Sheath weapon (interaction), retrieve kit (action)
2. Use kit (action), stow kit (interaction)
3. Draw weapon (interaction), attack or whatever (action)

Or some combination.

Sometimes you have to use reason to RP an interaction.  It seems to me a healer's kit is a First Aid kit.  You put your weapon down.  You pull out the kit.  You might use one hand to grab a 15th century gauze and apply pressure to a heavily bleeding wound and if the player is conscious they could maintain the pressure and you could pick up your weapon and be back in the fight.  The kit can just sit there next to the victim.  If it requires dressing a wound because the victim is unconscious, you'll need two hands.  Again, you can leave the kit on the floor until the battle is over.  Is the victim going anywhere?

I always carry my Healer's Kit attached to my belt. That way, at worst, its an Interact with Object followed by a Action to use.

Homebrewed Ruling:

In my campaigns, I use the house rule that if you have the Healer feat, you can use a Healer's Kit as a bonus action.