Should I keep my pieces defended?

Is it best to keep my pieces defended by another piece most of the time?

@1
"Is it best to keep my pieces defended by another piece most of the time?"
++ Yes. Nimzovich heavily recommended overprotection. That means your weak and strong points are all defended one time more than they are attacked. The advantage is, that all of your defending pieces stay free to move without giving up the protection. You are also not forced to respond to an additional attack, as all your weak and strong points already are overprotected.

Very good

There is a quote from Kasparov (I cannot seem to find it at the moment) where he states that he always tried to have every piece defended at least twice as that limited tactical opportunities.

Even ones not attacked? Everything defended twice seems impossible.

Seems tough, that might be why Kasparov said he always tried to have every piece defended twice.

Yes. The more pieces you have undefended the more likely you are to be exposed to tactics.

Yes, preferably by pawns. You need to be careful when you have undefended pieces, or pieces that are only defended by another piece which isn't a pawn. Also, you should look for your opponents undefended pieces because you may have tactics available to you. Also pay special attention to pieces which are defended and attacked. You should also pay special attention when your pieces line up, or your pieces are on forkable squares (forkable by a pawn or knight). You should see those tactics coming from a distance. You should see the potential of them appearing from a distance. Pay special attention as to how your opponent can make checkmate threats against you as well.  Pay attention to X rays and stand offs. Pay attention to forcing moves (which are mostly, but not exclusively, checks, captures and attacks). Finally, you should also see all of these things for yourself, not just for your opponent.

However, many beginners when they hear this start putting their bishops on d3,e3,d6,e6 while they still have their corresponding pawn on d3, or e3, or d6, or e6. Don't play such a bishop move if you are blocking your own pawn (there are exceptions for this rule, but you will learn these exceptions as you go).