I played a rated game where my opponent had time trouble and just as the last seconds ticked off he typed in chat "TIME PLEASE!"
I'm also confused about take backs and whether it's rude to not accept them.
There are a gazillion threads about the subject. There is no absolute guideline to what is considered (im)polite in online chess, but let me present my own view on the matter.
With respect to your first point, you could have increased your opponent's time, or you could have not. Either way, both you and your opponent decided to play a game within a certain time frame. When any of you approach zero seconds, you both have the right to ask for extra time (it even happened to me once in an OTB tournament game, which I politely declined). Though, of course, none of you are obligated to actually add time.
Take backs are an entirely different story, even though none of the players can be obligated to take back a move. There are, I think, two possible events which precede a takeback offer.
Firstly, there is the case of the mouse slip event. Here it is obvious to both players that there was a mouse slip involved, and it shows good sportsmanship that one takes back the opponent's move. Though, one could argue that during an internet game both players tacitly agree that such unfortunate events could happen and neither of you can be blamed for not taking back the move. However, when the opponent claims that a mouse slip happened, and you are not sure whether your opponent simply blundered, your opponent cannot blame you for not taking back his move.
Secondly, there are obvious blunders or pre-moves which could be considered a blunder. When your opponent asks for a take back in this case, your opponent cannot claim bad sportmanship if you decline his take back offer. I would even consider it to be bad sportsmanship if your opponent asks for his move to be taken back - which was an obvious blunder - and claims bad sportsmanship when you do not take his move back.
Answer above is probably the best response on this issue.
As above - whether or not you give the takeback or time is only going to stick in two peoples mind. If you are feeling altruistic - do it. If you're not don't. For the most part it won't matter - a game lost is only one away from being back where you were and two from being better.
Psychologically, socially - I prefer to err on the side of altruism. You will get fleeced sometimes - and they should be shot but hey - you'll always know they gained the upper hand - underhandedly :(
I almost always wish my opponent 'Good luck!' on commencing any game, rated or not. In casual games, I'm very lenient with takebacks and adding time if they're asked for, seeing as it's casual and most casual games I play are either with friends or challengers looking to learn from a stronger player.
Rated games, on the other hand, are quite another story. I *might* add additional time (but ONLY if it's asked for, not out of my own volition) and I will acknowledge takebacks for obvious mouseslips. These are usually apparent and asked for almost immediately after the move's made. There's no real-world equivalent for this and you do not accidentally place your piece on an adjacent square when you clearly meant it to be on another one. This is perfectly excusable.
Takebacks for moves that don't appear to be mouseslips, however, are those that I'll decline in rated games, *especially* if you ask for a takeback for your move *after* I've made my own. It should not take my move for you to see the error of your own, and the touch-move rule is very much a real-world equivalent. Once you've made your move and let go of the piece, it's done and you have to deal with the consequences.
I am a horrible person when I play rated games, so I probably won't be the best to advice on chess etiquette, but I think that those take-backs and time adding actually hurt sport - since they make the result unnatural - and by natural I mean with all the human weaknesses we have. Also nothing forces anyone to just not play games without increment thus solving clock problems and avoiding getting flagged in drawn/won positions (yes I do that often in bullet [actually mostly], but when I create blitz/classic games myself I usually use increment). Also I do not consider bullet to be chess, but some sort of action game with chess elements.
Mouse slips are indeed real, and they happen. However, I got tired of blunders being "mouse slips" after my move, politely declining and then having players ragequit or rant at me in chat rather than finishing the game. So I simple disallow takebacks in rated games now. I can honestly say in the last year I have made 0 mouse slips, so I think outside of super fast clocks where players under time pressure get careless and then oops mouse slip, it's actually a very rare reality if a player is aware they are playing on the internet and do things to reduce mouse slips (square clicking for long drops for instance). I feel like many mouse slips are akin to the touch rule, where a player has touched and even moved the piece and is holding it with a finger triple checking if that's the right square and then slips. If you are square clicking there is no chance of that happening and you even have the ability to chose another piece entirely, so there's that advantage. Thus, I just don't do takebacks in rated play.
Generally I don't mind losing on clock. However, I had two won positions yesterday, and I missed my force mating sequence because my opponent kept being super chatty during my turn. They ended up holding, and then because I spent so much clock distractedly trying to find the win, I had a clock disadvantage in the absolutely likely to be drawn endgames, I couldn't resolve the game quick enough. Their play style immediately went from chatty slow play to silent blitz play at the very minute my clock fell below 2 minutes and they of course refused to draw. Then they told me not to be angry when I refused their rematch.
Different people all have different ideas about sportsmanship when it comes to internet play. I think the easiest way to define what is good sportsmanship is to simply ask, "how would I expect to be fairly treated in this situation if reversed?"
I think it's fair to allow one mouse slip but only if you press take back less than 5 seconds after making the move. Also take backs should not be encouraged if the responding move was already made. Mouse slips do happen to almost everyone in blitz, though apparently not to you.
I agree ^^ with not-cheating, some people just say that a mouse-slips happen even when they were just blunders.
not-cheating, half the reason I haven't had a legit mouse slip in the last year is because I don't much play fast clocks. If you read the whole post, I said obviously it happens under faster clocks. The thing is, is it really a "mouse slip," or is it you examining a move quickly and then accidentally letting go of the button before you decide, much like accidentally taking your finger off a piece when you aren't 100% sure that's where you want it? That was my point. I think legitimate mouse slips are far rarer than people let on, not that they aren't legitimate ever.