How does Sharing is Caring work?

Sharing is Caring, the ultimate skill of the Best Friends Forever skill tree, gives Deathtrap a copy of your shield.

Does it transfer over the special effects as well? Does his shield recharge if given enough time? Will Amp shields or Maylay shields boost his damage at all?

The answer is yes. The shield Deathtrap receives is an exact copy of the one you have equipped when he is summoned. This means he will also receive any bonus effects the shield provides, including bonus roid damage.

Here's what the skill's page on the Borderlands Wiki says:

Sharing is Caring is a Tier 6 Action Skill Upgrade in Gaige's Best Friends Forever skill tree. Learning this skill will cause Deathtrap to be summoned with an exact copy of whatever shield is equipped on a player, and will provide the same bonuses/effects. Included in these kinds of effects are buffs such as "Roid" damage found on Bandit Maylay Shields and the Love Thumper, allowing Deathtrap to start doing huge damage at the cost of protection.

I can only imagine Deathtrap + maylay shield would result in a hilariously absurd damage output. Do let me know how it goes if you decide to try it =P

According to tests done by user Zikel on the Sharing is Caring skill (originally posted in the /r/Borderlands subreddit):

Deathtrap does not benefit from the special effect of Amp Shields.

Spike, Nova, Roid and Booster work as you would expect.

Absorb shields do not give you ammo, I can't tell if they are protecting DT from bullets.

^{Source: Reddit /r/Borderlands post by user Zikel, "Many skill descriptions are misleading, don't tell the full story or straight up lie to you. I took it upon my self to test EVERYTHING and clear up the confusion. Here are the results."}