The cooldown on companion gifts

UPDATE:

Apparently at some point, a patch has been applied to the game that now allows companions affection to be negative. See other answers for details. (I haven't confirmed this or I'd put the details in this answer.)

ORIGINAL:

No. Your companion's Affection can not go below the initial starting value, indicating neutral.

If you're on an RP server, feel free to role play their hatred towards you by staging a fight with them yourself, then never summoning them again after you betray them and let them die in a fight you tell them to start as you run away laughing at their demise.

Otherwise, (seriously) the benefit of having high Affection with a particular companion increases their chances of crafting and mission successes, yields better quality results and opens up better gear.

This depends on the Crew Skill.

Gathering skills, (i.e. Scavenging, Slicing, Bioanalysis and Archaeology) all have nodes scattered around various planets which can be gathered from, as in other MMO's.

Mission skills (i.e. Diplomacy, Investigation, Treasure Hunting and Underworld Trading) do not. The only way to use them is with missions.

In general, even with Gathering skills, sending your spare crew members on missions will result in a much better yield of materials, along with being able to target the specific mats you need without entering zoned which you cannot handle or which are trivial to you.