Well first of all that's not exactly what an ADC is. There are 2 definitions for ADCs 1) Attack Damage Carries 2) Attack Damage Casters. The first of those definitions actually came from DotA and literally means 'Someone who you carry through the early game in hopes they'll be able to carry you through late game' the 'Carry' in the term is not "I'm supposed to play super aggressive and get kills" because yeah those are nice but honestly all you need is the farm. The support is only laning with you because their job is in teamfights and they really have nothing better to do than help carry you through early game. The second C 'Caster' is someone you actually want to gank for, because they aren't going to carry you through late game in the same aspects, they really aren't a right clicker, without their abilities. Ez and Lucian are AD Casters. Lucian's passive (which is the main reason he does well in AA trades) are proced by his abilities, and all of his range is also in his abilities. Ez's Q is really what wins him trades. They don't siege towers the way a Jinx or Miss Fortune would.
If you're playing an ADCaster, that's when the jungler should focus you, because you're like an AD Midlaner, you're someone with good damage through abilities and sometimes AA buffs (think Ori and Ziggs as AP equals here) but you're not dependent on items the way a Twitch. ADCarries are specifically dependent on items (same with APCarries like Kayle) while ADCasters have good damage in their skills and aren't as dependent on items.
If it is a Carry then everyone should ensure his/her farming phase and dmg output environment are good?
So the answer to that is yes/no. Yes the reason why you're in botlane is because you're by a roaming mid, a jungler, and toplane has a teleport. But if you're going to go HAM and stay pushed out and your positioning is already bad (because you're in position for repetitive ganks) don't expect the rest of your team to believe you're going to position yourself safely in teamfights. The hopes is that you carry them through late game, it's not a fact and the ADC who doesn't know how to position themselves safely in lane, is hyper aggressive, and keeps going aggro when they should just be farming once they zone the enemies out (unless the enemy is stupid and steps out of position themselves) that's not who the jungler/midlaner should be expecting to carry. That's the kind of person who goes into 1v1s late game because they think their job is to kill things. It's not. You just shoot the front line to provide some sustained damage, and siege towers. That's you're job. You carry the team because in order to win the game they need to take towers and you're the one who rips them down. If they can't expect you to play nice and rip towers down, the jungler isn't going to focus your lane.
So no your damage output is already good, and your farming phase was already good. They didn't need to help you ensure those were good. You just need to back off a little and farm. And by that I mean Zyra's snare shouldn't be on CD so often, it should be up for when you need it. You should have wards up so that when you see the jungler, Zyra can snare them and you can back off. You guys have escapes. If you guys can't avoid ganks it's because you're being super aggro by using your escapes, and not controlling the lane the way you should. If you have them so broken down that they need a jungler to do anything in your lane, have them zoned out and freeze the lane at your side of the river so that they put themselves in position to be ganked, and so you're safe. When the enemy jungler is ganking it's because you're pushed out and your escapes are on CD.
It is like they don't think it is a good idea to counter gank a high damage/low escape lane.
And to sum it up, if that high damage/low escape lane is always out of position, then they shouldn't expect you to be able to position yourself in teamfights to live long enough to do the damage. So no they don't think it's a good idea to counter gank you.