How can I mitigate some of the downtime between player's turns?
I think the main points have mostly been hit. A few strategies: 


Direct interaction during other players' turns. For example, if there is a combat element, if the defending player has interesting choices such as playing cards from hand to gain advantage. As mentioned, Magic is a good example as the defending player has plenty of decisions to make and cards to play. Counterpoint is Risk, where defender just rolls dice doesn't work as well for keeping interest.
Indirect interaction during other players' turns. In a recent game Eminent Domain, player gets to piggyback on the decision the current player makes. Or in a game like Battlestar Galatica, there is so much meta-game discussion that regardless of whose turn it is, all players are engaged and invested even if the game mechanism doesn't allow them to interact with the "game" at that moment.
Pre-planning while avoiding Analysis Paralysis. Player can more or less make all decisions about their turn while other players are doing their own. A key part of this is avoiding adding new information during current player's turn. For example, if the player must draw a card, have them draw at end of turn rather than beginning of turn. This will give them more time to internalize their choices.
If there really is no way to do any of the above, then you need to find a balance in length of turn. Perhaps make really short turns, but if the player only has a few actions each turn they may be all the more frustrated because they feel like they can't get anything done on their turn. If you instead give them lots of actions, you can make the player's turn feel really beefy, but that's going to make the wait for next go-around that much worse.


I don't think there's a golden rule for amount of downtime. Instead, it's all about perceived downtime. My "turn" may only come up every 15 minutes, but as long as I am thoroughly engaged in what my opponents are doing during their turns, and I have true decisions and actions to take during their turns... it doesn't matter.