I know there's only 2 of multi-level all the game.
And I know it allows only 1 tab in 1 browser.
Can I play my current 3 maps (used 2 multi-level and 2 skip-level) in 3 browsers? And will this affect game saving?

Another question. Since:
1. there's only 1 set of artifacts, and
2. according to the popup when switching game, the idle bonus are calculated when entering the map.
Is this a good way: when switching to a map that has been idled for 2:15, equip a "idle time" card before switching map, to get 2:20 bonus in the opened map?

You can use different browsers, but never at the same time. You must close idle completely in one before opening it on another. See https://www.warzone.com/wiki/Warzone_Idle_Syncing

Just go to menu, change map, play it, repeat :D