Actually you want a course in thermodynamics summarized. I'm not sure you will get it here. I hope others will add their thoughts/ Perhaps enough comment will get it across to you.
First, ALLreactions , given enough time, will reach equilibrium. It isn't that some do and some don't. All do but thermodynamics doesn't say anything about how long that may take. Also some reactions are at equilibrium with just 1% of the products formed while others are at equilibrium when 99.999% products are formed. The value of Keq are random; i.e., they vary all over the map depending upon the reaction under investigation. However, Keq is Keq and there are no exceptions to that value as long as all of the conditions are met.. The pattern is Keq expression which I won't write here. As to the entropy thing, my thermo prof drilled it into us that our universe tends to disorder; i.e., it doesn't like symmetry and everything organized just so so; therefore, reactions that will produce more disorder are spontaneous. This is off the cuff but perhaps it will help.
Can someone please explain why reactions go to equilibrium? Does it have something to do with entropy? Are k values random and full of exceptions, or is there some pattern behind it? What makes something thermodynamically favorable in terms on entropy? I know the entropy of a system and surroundings must be positive, but why does that being positive guarantee that the reaction or process is favored?
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