[RPG] Can you take a “free object interaction” while incapacitated

Unfortunately, it would require your action. You can sheath a weapon or draw a weapon using your free object interaction, but not both. (Addendum: The Dual Wielder feat allows you to draw 2 weapons or sheath 2 weapons using a single object interaction. Note that this does nothing to fix your problem, it just seemed worth mentioning.)

A Rogue with the Thief archetype is absolutely free to use their Cunning Action to Use An Object on any object they choose (assuming they have the feature, obviously), and the scenario you describe is a perfectly legitimate use for Cunning Action.

However, there is another solution for all the non-Rogue players who want to switch weapons and attack in a single turn: Drop your weapon, don't sheathe it.

Dropping something requires no action at all, not even your free object interaction. So you can drop your weapon, use your free object interaction to draw a different weapon, and attack with that weapon on a single turn. And move too, if you want. Just remember to come pick up your dropped weapon later (probably at the end of the combat.)

No, the ring of free action doesn't prevent being incapacitated

The incapacitated condition is separate from the restrained and paralyzed conditions (it being inflicted by the latter is irrelevant here) and is not a reduction to speed.^† Thus, neither of these effects give immunity to being incapacitated.

If the ring had given immunity to the incapacitated condition, it would have said so (probably by listing it among the other mentioned conditions).

† The incapacitated condition is inflicted by some other conditions (paralyzed, stunned, unconscious) which do prevent movement. However the full effect of the incapacitated condition alone is (per Appendix A of the PHB/Basic Rules):

• An incapacitated creature can't take actions or reactions.

In 5e, movement is wholly separate from actions (and reactions).