How do i convince split pushers to group?
You often can't (even in higher ELOs). Just make sure you have disengage/waveclear to prevent the enemy team from capitalizing long enough for the splitpusher to threaten something.
How do I convince people to make a formation and peel in general?
Aside from delivering specific instructions (i.e. "CC the Akali when she jumps on me") and hoping they do so, you can't. Generally, I would just stand behind/beside your tanks or supports; they will generally target someone nearby so you use them to do so.
Win lane lose game is my most common scenario: How do I prioritize when and where to carry that lead to other lanes?
Situational. Generally, I would try to recognize who else is doing well (i.e. which lanes are winning or seem competent) and help them out.
How do you help a feeding bot lane?
Still haven't figured that out considering how badly that lane can snowball. Generally, I would prioritize an ally capable of killing the enemy ADC rather than try to recover our own. Kill the enemy bottom lane when you see the chance, but don't force anything or risk a counter-gank.
Itemization when behind: how do you know when to go visage vs banshee or randuin's vs frozen H. etc?
Situational.. Depends on team compositions and in-game situation.
Generally, I recommend Silver players looking to climb grab someone with good poke, safe waveclear and some CC.
Last season, I found most Silver games would devolve into both teams grouping around a random objective and having a standoff until someone was caught out. The ability to stall pushes and poke down the enemy team before a fight starts is fantastic. If you also have the ability to (dis)engage fights, that is even better.
Last season, I used Varus (ADC) and Ziggs (Mid) to climb out of Silver since they worked well in that environment. Nowadays, I use Lux for similar reasons.