Aristotle in his Poetics examines three kinds of poetry - comedy, the epic & tragedy; he says
Since the objects of imitation (mimesis) are men in action, and these men must be either of a higher or a lower type (for moral character mainly answers to these divisions, goodness and badness being the distinguishing marks of moral differences), it follows that we must represent men either as better than in real life, or as worse, or as they are
By imitation, he does not mean mere imitation, but a condensation; that in comedy is caricature; and in tragedy and the epic aims at elevation and grandeur.
He notes that:
It is the same in painting.
For example the caricatures of a Cruickshank, or Grosz; grandeur in the history paintings of Caravaggio or De La Tour.
for Comedy aims at representing men as worse, Tragedy as better than in actual life.
For life itself being the measure or the standard; art aims either above or below; pure imitation will be as a mirror and adds nothing new; thus again imitation as condensation.
and he repeats:
The graver spirits imitated noble actions, and the actions of good men. The more trivial sort imitated the actions of meaner persons, at first composing satires, as the former did hymns to the gods and the praises of famous men.
Hence the laughter provoked by comedy, according to Aristotle, is not that which provokes our pity; but where we recognise meanness or lowness of spirit.
As for Comedy, it is an imitation of men worse than the average; worse, however, not as regards any and every sort of fault, but only as regards one particular kind, the Ridiculous, which is a species of the Ugly.
This sets it of from Nobleness which is a species of Beauty. He expands:
It consists in some defect or ugliness which is not painful or destructive. To take an obvious example, the comic mask is ugly and distorted, but does not imply pain.
If it was painful or destructive, in a proper sense, then it would be a species of Horror; and this would jar against the good humour of comedy.
That which provokes our pity (one could say here, the pitiful; but one shouldn't because its sense is no longer consonant with that which provokes pity; but like the word pathetic, which was originally meant that which had pathos; now both are close to meaning that which deserves our contempt; this movement in emotional value is of significance itself; but I digress) according to Aristotle, is the emotion proper to tragedy; hence he says:
For the plot ought to be so constructed that, even without the aid of the eye, he who hears the tale told will thrill with horror and melt to pity at what takes place.
Thus Oedipus Rex, Iphigenia in Aulis or Medea.
It is the arousal of pity and horror, and then their cleansing or purification that Aristotle calls Cartharsis. To now go onto the points that you raise:
In a crude sense, I suppose he means we laugh at the pitiful;
He means that it reveals the crudeness of both the audience and the comedian; but there are degrees of comedy; he notes for example comedy in Homer - whichis an epic.
I've been guilty of it and this kind of humour is "sick," but I think Aristotle is right. Should one feel bad because they laugh at the misfortunes of others?
Theres an implicit value judgement in Aristotles Poetics which judges the higher better or nobler than the lower; he wouldn't say that it is'sick' but that it is 'low'; and thus the appropriate response is laughter; but to laugh at misfortune proper; the kind of mis-fortune that dogged Oedipus is, according to Aristotle, the wrong response; and thus 'sick'.