There are two ways to talk to them in private: in cantinas and in your personal ship. You'll know by their portrait when a character wants to talk in private, because an icon will appear in it. Indeed, the codex gives you an idea what these companions like and dislike. The companion's affection is based on the choices you make in dialog and on the gifts you decide on giving them.
And yes, you can equip companions with new gear and upgrades. The way to do this when you acquire gear (which may or may not be specifically for a certain companion) is to open the character screen (use the key 'c') and open the tab of the companion and equip the gear you want by right clicking on it in the inventory (key 'i') screen. Customizations work in the same way.
A companion cannot revive you. When you die, the companion dies with you, even though you can still fight when your companion is defeated. You cannot revive anyone while in combat anyway. And yes, you select your companion in the crew window ('n' key). There's a button for that on the right in that screen. You can only have one active companion, and every class has six companions (including the droid that comes with the ship).