Would posting comment (e.g thanks) increase your chance to win the giveaway?

Technically no, but it's polite.

No, but posting this thread lowers your chance to be taken seriously though.

No, it doesn't increase your chances and it's just there as a nice little statistic you can keep track of. Most private giveaway groups do keep them in account when recruiting new members on the forum here though, so if you would ever like to get into one of those it's always a plus to have a good balanced entries to comments ratio. Though I have to admit I also didn't really comment in my early days of SteamGifts, I steered clear from that path now though! :D

But the most important part, as said before: It's polite to do so and makes the giveaway creator feel like his contribution is appreciated!

Posting comments doesn't directly effect your chances in a giveaway.

A user that is active in the community however (posting in topics and making giveaways) is more likely to be noticed and invited into private giveaway groups, which will boost the chances of a win.

Also, read this

It doesn't boost your chances but the GA creator will see, you are not just entering mindlessly in GA-s but you spend a little time to thank the GA creator.

Not unless you mean it.

If a private or group giveaway asks you to post thanks and it's accepted by support, if you don't and you win then the giveaway will be rerolled.