Are supports supposed to be behind the adc or in front of the adc?

depends on the support

Depends on the class. Enchanters in the back, wardens in the front, vanguards in the middle of the enemy team. (Of course with some exceptions)

Supports are supposed to CS and be the primary damage source for the team

Which support? Some stand behind or at the same level and some generally play forward. It really depends on the lane match-up and the game situation as well. There are a lot of variables to this honestly.

Yes. Both. And neither. Most of the time on most supports you want to be even. If you're in a disadvantage against engage you want your support to be slightly behind to make disengage threats, if you have the advantage you'll usually be pressuring forward, unless you're under jungle pressure. It's also very supp dependent, as ornn said.

Some have to stay in the back. You can't expect Sona or Soraka to be the frontliners.

Case changes if you have a Leona or something like Nautilus in lane.

And then there are supports that play alongside you, like Morgana (disgusting new look aside) or Lulu

Generally they should be in the same line for zoning purposes. It of course depends on the lane match up.

Depends who you are against, or what you play...for example, if the enemy has a blitz, and you have a shen, well, block the hook for your adc, don't just stay in the back ;)...

Depends. Usually in lane being parallel with adc is ideal. Outside of lane phase it all depends on what kind of support it is and what kind of team you’re facing. If it’s a warden type support you will probably hover around carries. If it is a catcher probably will be closer to frontline. If it’s an enchanter probably want to stick close to carries again.

It really depends on the support champ that you're playing with. If you are playing a support champ like Blitzcrank, then I usually am in front of my ADC to land hooks and then knock them up. Other support champs that I play that I consider squishy, such as Yuumi, Lux, and Lulu, I usually play in the backline. Overall, it comes down to what support champ your support is playing.

generally they are supposed to be even......that way, if your adc gets engaged on...your support should enage on their adc......

usually, you try to stay in a sort of triangle formation, so can make little 2v1 trades, so parallel to the adc. But to be fair it´s pretty champion specific.

as nami, soraka I play fairly aggressive and stay in front of my adc, while on taric I often stay behind my adc which sounds odd since taric is a tanky support, but the matter of the fact is that you don´t really have a lot of range to work with and that your E is better applied defensively than offensively due to how telegraphed it is. With that said, you would just get poked to death without accomplishing anything, so it´s often better to just stay behind, bait with your adc and turn the fight with a well-placed E.

It depends on the champ, matchup, time, amount of vision, the presumed position of enemy jglers etc.

Don't understand why this got downvoted; very good question.

If it's an enchanter, you're supposed to position just in back to provide ADCs heals and shields. If it's a defensive tank, you position in the front to protect your carry. Offensive tanks and carry supports should position themselves forward and toward the brush to flank and protect against enemy flanks if needed. In the current meta, Lux is an exception to this, since she's being played more like an enchanter than a carry support, and should probably position just in back of the carry.