Does increasing the entropy of a system cause it to absorb heat?
I consider watching any video a waste of time, so I'll be judging from your words alone. (Anyway, your question is essentially self-consistent, which is good.)


Yes, entropy is a measure of disorder (sort of).
No, $dS={\delta Q\over T}$ only for reversible processes.
And no, entropy is not the Q-T ratio for two reasons. First, because the above formula deals with infinitesimal increments and not with finite quantities. Second, because like I just said, that formula is not universally applicable.


All in all, the increase of entropy alone is not necessarily accompanied by the heat flow to the system. It may be so, but it may be otherwise. You may imagine pretty much any kind of possible outcome:


You dissolve some $\ce{NH4NO3}$ in water (which surely increases entropy), and the beaker gets palpably cold. The system starts sucking heat from the surroundings, just like you anticipated.
You mix your $\rm^3He$ with $\rm^4He$, and... nothing happens (except the increase in entropy, of course, but that's a thing you can't see or touch). There are no thermal effects to speak about, and hence no heat flow.
You burn a candle in a closed vessel (supposedly, containing enough air), and guess what? Entropy increased, but heat is flowing outside.


As for your last remark: true, it takes some energy to decrease entropy locally (say, in your fridge). But in doing so, we inevitably increase entropy elsewhere, so the overall balance is positive.

So it goes.