Do all 5 cards attack the other player, inflicting 7 damage, and then becoming discarded?
No, cards in your hand are not yet in play and thus cannot attack.
What about playing one of the JJ McTeagues, discarding the other four cards, would that then give me six points of influence to gain more cards?
Yes, just keep in mind how chaining works: if you start the chain with J.J. you can then play another red card (small moon color) or pass. If you pass then your opponent has the chance to play a red card. When J.J. resolves you'll have to target a minion to damage, so make sure you're not the only one with minions in play at that time!
The instructions state "if you have any minions on play at the start of your turn you must use them" but I don't understand how one can have anything in play before a turn actually starts.
The first player won't have any minions in play on their first turn as they haven't yet had the chance to chain them out. Everyone else however can potentially attack on their first turn if they managed to chain cards on the first player's chain.
How do cards target other player's cards?
Effects like J.J.'s who actually say "target" in the text have their targets chosen when they resolve on the chain. When you attack in combat you don't attack individual minions but rather you attack the player and then the defending player chooses which of their minions block each of your attackers.
Are there cards that the other player will leave face up?
The only cards left face up are: archives (the stacks of cards for purchasing from), discard piles and any minions that have been placed into play.
Hope that helps