Mid advice - League of Legends

I main mid/jungle and I'd say I play in lower Gold level atm. 

To your first point i'd say that sometimes lack of farming can't be helped if you are getting kills from roaming. If you're in your own lane getting kills there should be no reason that you get behind in farm. The death of your opponant should, in most cases, allow you a wave of farm...and if you had advantage you are often strong enough to deny some farm or mentally pressure your opponant to maintain a steady lead in cs.

Secondly, about following lane opponants to bot lane, it can be a little tricky...People will always say you HAVE to follow, and i think thats true. There are some cases where it may not be a good idea, but bad plays can be avoided by having a ward on blue sides bush at the crossroad near river and wraiths. The only reason i ever get hesitant to follow is if that bush has no vision, they could easily anticipate me following and jump me blindly otherwise. I'd say always try to keep that warded. On top of it keeping you safe it also gives a HUGE forewarning to bot lane when your lane opponant roams.

For your third point i'd like to say it depends on what kind of plays you could make if you decide to leave your turret. what can you gain by leaving or roaming vs what kind of gain does the enemy laner, anivia in this case, gain? For this situation i'd say make a good shove of that lane and bolt for bot lane. Either the anivia will follow you, leaving your tower at no risk of falling...or she'll push lane and take tower. In your situation you were 4/1 and she was 1/4, probably similar farm if not you being farther ahead. In this case you already have the advantage on her even if she takes tower and on top of that she wont really catch up by much if she does take the tower. If you can make roaming count...ganking or diving bot lane and taking turret for example, it makes it WAY easier to take mid tower when returning to lane. 

Alternatively you can coordinate a gank mid and just try to kill anivia if its a farmfest. this would ultimately free up the lane to push hard and take tower. I find this a less viable option though against a great wave clearer like anivia. Overall I'd say if you would rather stick it out in lane definitely play it safer in this situation, you already have the advantage and sooner or later both towers will be down eventually. later down the road if you hold advantage you will be stronger than her in teamfights.