how accurate are the analysis options?

I have found them useful. Presumably, the "deeper" the analysis the more thorough the information provided. I always review the analysis after each game to see both the suggested moves for both myself and opponent. I try to see why alternatives are suggested and try to remember for future games. I don't "study" the analysis, just review. I would likely get more out of it by really picking apart the game afterward.

So, I think it's reliable and useful.

Short answer: they're very accurate, but not perfect. 

Slightly longer answer: they won't miss tactical shots, but their positional judgement can be challenged. And, the engine may make some endgame mistakes that a human can catch.

But in general, you're getting +2900 level analysis. That's not too shabby.

I always like to choose the "Self-Analysis" option after the game is complete, which brings up a new window and has Stockfish analyze the position. You can see how it evaluates several different moves, and have it think as long as you want on critical moves that happened during the game. I like to go back and play thru the entire game using this option, and play alternate moves that I was considering in certain positions. It gives you a lot more information than the other analyze options. 

Quick analysis works fine for me.  But sometimes I also use self analysis.  Self analysis allows you to go through the game move by move, and shows the different options for best moves.  

So for example if you play 7 - Nd5, quick analysis shows you if it was a bad move, and says nothing about it if it was not determined to be bad (not bad enough to fit its definition of an inaccuracy).  

Self analysis gives you a numerical result for Nd5 and also shows what the better moves were, with their numerical scores as well.  

So normally I am bad enough that I look at quick analysis an see BLUNDER, you could have had mate in 1!!!  or something like that, and knowing the big mistakes is enough for me.  But If I am really interested in a certain position, then self analysis is better for seeing the best ways to play it.  Better than that, I download the game and analyze it with Lucas chess (free software).  

Deep analysis looks a lot like quick analysis, but slower.  I paid to be premium so I could use it, but don't really see a need to use it.  

It serves my purposes just fine:  it blunder-checks.  

Why would you want an engine that blunders check?

Its pretty good for building your chess brain, but there is some flaws I noticed. I played someone who blundered his Queen away plus a rook. I played the position and ran it through an engine and it was a blunder on the analysis part. Its rare but when you do see it make a mistake its pretty obvious and very rare.

The analysis, even the quick analysis is pretty good as a blunder check. Beyond that, it is pretty limited. For example the CAPS scoring as currently implemented is pretty worthless. Say you get into an endgame where you have many different ways of winning. The computer likes a specific way and your CAPS score is likely to get hammered as a result. In essence, you get penalized for getting into an overwhelmingly good position. Or alternatively, you end up in an ending the is pretty simple with many fairly forced moves. Your CAPS score skyrockets.

In real games with longer time frames, blunders often occur because of mistakes that occurred earlier that put you in a situation that was hard for a human to hold. Likewise, wins occur because you gradually put more and more pressure on a position until your opponent cracked. The analysis is not good at identifying either of these situations and opportunities to get into/avoid them.

And analysis cannot identify positions that you are comfortable with, vs uncomfortable. For example you may have a choice of taking a captures which transforms a game from Rook and Pawns to King and Pawns. There may be little difference from the engines point of view, which you enter into. But from a practical point of view, it may make a huge difference. King and Pawns endings can be notoriously difficult. Rook and Pawns can as a practical matter be easier to hold.

An engine doesn't care that you avoided a move because you know you are walking into a line that your opponent is probably very well booked in.

My basic point is that engine analysis has its place, but there is still no substitute for doing it yourself.

I believe the algorithm has a weighted proclivity toward reducing material, which at a rank of 1180, am not qualified to say whether that’s the way to go; however, in doing so, it may appear it leads you into some weaker positions, which I avoid.  The latter may explain why am only 1180.