The copies are put directly onto the stack. They aren't cast and won't be counted by other spells with storm cast later in the turn.

A copy of a spell can be countered like any other spell, but each copy must be countered individually. Countering a spell with storm won't affect the copies.

The triggered ability that creates the copies can itself be countered by anything that can counter a triggered ability. If it is countered, no copies will be put onto the stack.

Time counters are usually found on cards with suspend and vanishing, but may be found on other cards as well. Notably, Sagas use lore counters to track their progress, not time counters. You can't move a Saga's chapters forward and backward this way.

Spells cast from zones other than a player's hand and spells that were countered are counted by the storm ability.

To time travel, look at each permanent you control with a time counter on it and each suspended card you own in exile with a time counter on it. For each of them, you choose whether you want to put a time counter on that card or permanent, remove a time counter from it, or do neither. Then those changes all happen simultaneously.