Every character in ASOIAF is misogynistic. He killed someone who aimed to kill him AND orchestrated the rape of the love of his life. He has shown quite a bit of concern for the common folk, and was one of the first noble characters to do so.
I don't remember the sex slave, so I'll have to look that up. But Dance of Dragon is a meandering tale of him coming out of his drunken depression and finding new purpose. Keep in mind that he knows Cersei has people going around killing dwarves looking for him...
Tyrion had justifiable reasons for killing Tywin.
But Tyrion has also fed a singer to the poor peasants.
He killed Shae slowly out of rage for feeling humiliated by her for that giant of Lannister comment. GRRM even calls it a much blacker deed than what he did to Tywin.
Tyrion doesn't really care about the smallfolk. His reply when hearing about villages being raped, pillaged and razed is "So what? That's war."
Tyrion also thinks democracy is stupid and that giving a voice to women is ludicrous
Tyrion's arc is about becoming a villain but also about getting out of his depression.
There's a TWOW sample chapter where Tyrion almost murders Penny after an innocent comment by her