Does report abuse actually work?

I have attached this screenshot of how a player in the last seconds of the game abused and won since I could not focus being a newbie and got disturbed by this message. I had the piece and a little bit of time advantage before he/she sent this. 

After the game, I reported the player but the point is - this is nothing new, and I keep wondering if there is really any moderation that happens by the website admins on such issues. I understand this is a fairly common problem but still its rampant across many games. 

Secondly, is there any way if an opponent starts abusing/trolling one can stop the game midway and block the person without losing any ratings?

Yes.

No.

In home -> settings -> live chess -> game chat

you can set it so that chat is disabled automatically at the start of the game. Maybe you will prefer that?

keep in mind that a good couple million people play a day. if 500,000 people act up, and the chess.com team has only a few people moderating that day, of course it's gonna take a while. eventually you'll get a message from chess.com support saying that they reviewed your report and are either going to act or not. usually they act. it's rare for them not to.

I have always had excellent response on reports I've filed.

I've had a number of horrible messages on here and nothing seems to get done after I've reported them. Then I laugh when chess.com asks me to pay and subscribe. Lol, not until they clean this platform up

Not sure I’ve reported a few players but one in particular was someone telling me to “kys” and it’s one of the main reasons I started playing was to not. Someone like that should get hardware perma ban 

The Urban Dictionary says that could mean "Keep yourself safe."

A perma ban seems harsh.

I am sorry to hear that you have had people abuse you but you probably wouldn't know what action is taken. 

I've reported multiple people for quitting/stalling (it happens all the time - a major downside of chess.com) and haven't received a reply for at least a year, maybe two. The growth of the website has seen a degradation of the service - perhaps you should hire more moderators with your increased revenue?

The site stopped sending any replies to reports made through the report process quite a while back. They also have added a lot of support staff in the last six months.