As I see no high elo adc answered yet I will bestow my unranked game knowledge on you.
Before all, I highly recommend you to go Vayne mid (PTA with the 100 hp and biscuits, rush hex vs good ap, rush tabbies vs talon) if you can dodge skill shots.
Double adc team comp especially with Vayne state vs mages right now is really good and I have great success with it (op).
I often find myself split pushing a lane while my team is engaging in team fights and losing when outnumbered
Yeah it used to happen to me a lot.. as a late game scaling adc I need to farm till I have somewhat of a power spike but my team just kept on ARAM it, even if they didn't lose they got trades and a 2/0/1 kog maw is weaker than a 10/16/0 kog, so you are getting more and more behind.
When I see that a fight is going to happen mid If I think I can put some dmg - I just make sure the wave will get under their turret by the time the teamfight is over (so killing one wave) and rushing to mid, yes you are losing farm but you do dmg so you are helping your team in that team-fight and getting those trades are good for you, meanwhile the wave is pushing towards them so if you win the team-fight you can take a turret.
Try and ask them to only wait 30 sec for you to push the wave and join the fight, usually they won't listen, but try.. spam ping and "plzzz".
To be effective in those ARAM style games you need a champion that can perform ok in early but still strong in late cause low elo games get dragged.
You also need a champion that relatively self efficient (not needing the team too much for peel while still putting dmg).
mf, ez, jhin are in the meta right now (not sure about jhin tho) they are strong early but in late game they are practically a minions to a Vayne and propably to the rest of the crit based champs, i suggest you to try 
as starters (don't get tempted by ez in my opinion he is a really (really) bad adc to start with, try some games but dont main him) (I really likes tris, her passive and w makes her safe, her objective control is really good and she isn't so weak early)
Do I just follow my tanks around?
You never face check and in fights usually always behind one, never tempt to stand before them and poke or initiate, just remind them they are the front line and they should go and tank for you so you can poke.
Should I be split pushing?
Split push is usually for champions which can 1v2 or have good escapes, you are a major dmg source in team-fights late game, without you your team will lose to their adc dmg, you cannot afford to take the risk.
If you know it safe to push do it (the enemy dead and you see the rest on the map far from you), you will get more cs and the wave will crash on the turret making them to lose some cs and be far away from you while you taking objective like drag or mid-lane.
The more important question is "How to decide when its a good time to farm"
What do I do when my team doesn't listen to ping objectives
PING SPAM LIKE CRAZY and btch about it in chat.
Try to tell them before hand what is the short term plane, for example - "ww I see you just based and you have your botside jungle on, we will freeze the lane in our side and try to poke trade a little now and than you come a gank and we can drag after it"
What can I do to carry games if I cant get a 5v5 team fight engaged and people get picked off constantly?
Buy pink, ask them to buy pink, every time one of them have ward up ping them where to put it, just keep nagging them about wards basically.
team fighting tip that sometimes works-
If your mechanics need to improve and you in the point you do a good amounts of dmg (late game for example or IE + zeal even) you maybe want to join the team-fight like 3 to 5 sec after it started with a flank - most likely your team won't peel for you and if you are there in the beginning of the fight the enemy team will hard focus you, but if you join little late they are on cd, they tunnel vision on someone and you can start and creep your way to their adc kill him and severely injure the dps mage near him.
Personally I think 
mid in low elos is the best, that champ is strong early, strong late, have good split push, good wave clear, easy to farm with, fast turrets, good gank opportunities (push the wave fast and can roam botside, if the have knock-up the gank is really easy)
or a mage support, it is in your best interrest to start with shield instead of blade.
