Should you always try your best (play as if your opponent is a grandmaster)? Should you always play with your "best" skills no matter how (un)skilled your opponent is?

Is there a situation where intentionally not playing to the best of your abilities is a good strategy? Like, let's say you're playing against a chess noob. You don't need to waste moves defending your army because your opponent probably won't even notice the hanging pieces. You can checkmate your opponent faster by doing things that are usually considered bad strategies, such as moving your queen out too early in the game.

Additionally, I saw in the Chess.com mobile app that its "best move" hints change relative to the computer opponent's level. Seems like you should play against newbies differently than how you play against grandmasters.

Is purposely "not trying your best" sometimes advantageous?
Is it a good strategy to always play like a grandmaster?

Obviously yes, but unfortunately 99% of chess players are unable to do this even if they wanted to. ;)

Is it a good strategy to knowingly play moves that you know (or strongly suspect) to be mistakes?

No. 

"Mistake" in the sense that there is an answer that screws you if your opponent finds it. 
Because either they find it (short term downside) or it works out and you just reinforced a flawed pattern in your brain (long term downside). Blundering your pieces left and right is not a sound strategy, not even against "noobs" (after all, you still need to have some material left to actually mate).

Is it a good strategy to knowingly play moves that you know (or strongly suspect) to not be best (but still okay)?

Sometimes. 

In many situations, there are multiple candidate moves and it may have some benefit to play the move you judge second or third instead of the "best" one. 

Sometimes it's better to settle for a small but safe strategical advantage (that you are confident to transform into a win against a "noob" anyways) instead of a potentially big advantage that needs a lot of precise tactical play to hold (with the additional risk that you miscalculated somewhere). 

Keeping the game less complicated when ahead gives the better player some meta advantage (he's less likely to blunder, and as long as he doesn't, he will eventually win), which might make an otherwise suboptimal move the most advantageous.

Or vice versa, sometimes it may be worth it to go for a move that gives your opponent some advantage in one aspect of the game, while you only get a smaller one in some other aspect. Most gambits come to mind, for example. What matters is that you still should not play clear mistakes, only slightly "worse" moves where you do get some compensation. Sacrificing a pawn for initiative that's probably worth less than a pawn, or even sacrificing a piece for a mate attack is fine. Sacrificing initiative by moving out the queen too early for some hit-or-miss chance to apply the Scholar's mate is not a good strategy, because if they find a good answer, you end up with no compensation at all; and if they don't, they are so inexperienced that any healthy opening would probably have won easily (and quickly!) as well without using a strategy that is useless against better players (again: either short or long term disadvantage).

Making the game more complicated when things are even gives the better player some meta advantage (he's more likely to find a better way through the jungle), which might make an otherwise suboptimal move the most advantageous.

However, these kinds of strategies are used by grandmasters as well, so ironically you are still playing like them.

Is it a good strategy to knowingly play moves that you are not sure about (they may be good, they may be bad)?

No. 

Often, this cannot be avoided, of course (your skill sets limits to calculation depth and positional judgment). Thus everyone has to do it to some degree. It's not a "good" strategy though. Ideally you should always try your best to evaluate a given position.