Can "Good Luck" and "Good Game" be taken as insults?

Of course, in real life what matters is whether he was annoying enough to punch in the nose, knock to the floor, and drop crossovers on his face with your knee on his chest.

If it's not, there's no need to play all girlie and give it more time to worry about than the half-second 'incident' took up in the first place.

Quit crying about what the bad man did to you!

Sometimes playing this game is positively embarrassing.

I never say good luck, because really the only way my opponent can get good luck is if I blunder.  

Its dishonest.  I can hope we both play well but I really don't want to blunder. 

I try to take it positively. The majority of the time, I honestly think most are just trying to be friendly. Responding politely to rude comments is my own attempt to rise above their behavior and no matter how rude a player is to me, I will always always say "thank you for the game". I hope it's just saying "you didn't get to me" although, truth be told, the rudeness does get to me at times. For that reason, I generally don't play random people and only play others whom I know from mutual groups, etc.

Because my rating is so low, if I know the other player has a sense of humor, if s/he says to me, "good luck!", I'll often say, "thanks! I need it! ;)". I try to find humor in most things when possible. Usually, though, I take both good luck and good game (even if I sucked and lost in nine moves -- it's happened!) as gestures of good sportsmanship although I'll admit to saying "well, no it wasn't! But thank you for the game".

All I can try to do is hopefully try to set a good example albeit we are all human!

Thank you.

It is not "dishonest" to say "good luck".  It is a friendly greetings.

And you are very wrong when you say "the only way my opponent can get good luck is if I blunder." There are many ways your opponent can get good luck. He might learn he won the lottery while playing you.  He might be lower rated and hold you to a draw.  His headache medicine might kick in while playing you. He might have the good luck of you showing some kind of opening he does not know but is interested in. Luck does not revolve around you might make a blunder. 

Well put. A little humility and good-naturedness can even defuse a cocky, arrogant jerk who just doesn't realize how bad they sound.

I always try to give a "good game" after any game, win or loss. It is good sportsmanship.

I always say, "good luck" before the game because luck has no part. I say, "good game" when it was well-played on both sides and a little more often when I lose.

I whish them "eine spannende Partie" or "viel Spaß" at the beginning, when shaking hands. Wishing "good luck" is quite stupid because this means you wish yourself a blunder. Be careful what you wish, sometimes your wishes come true. Don't know what's wrong with "good game" though.

I always say, "good morning" at all times of the day. It is always morning somewhere.

I beg to differ with Estragon. The implication of saying "good luck" is that "they're going to need it." Why would you wish someone good luck if they DIDN'T need it?

Actually it would be more courteous to say "wish me good luck" at the beginning of a game, showing respect to your opponent.

People say like I'm going down and then I get them in Scholar's mate and they call me (insert vulgar word of your choice)

Not to get too carried away with this as I agree "good luck" can just be a greeting and a normal polite way to begin any game.  

But...

If he holds me to a draw even though he is a lower rated player then either I made a mistake (which would be his luck) or he just played a very good game (In which case I wouldn't say he was "lucky" as that might be mildly insulting. Eg., Carlsen didn't say Anand was lucky when he drew some games.)

it reminds me of a joke. two psychiatrists pass each other on the sidewalk. One says to the other 'good day'. The other thinks to himself 'I wonder what he meant by that?'. 

More polite forms one could use would be "have an enjoyable game" and "handshake" but most of the time nothing needs to be said.

Most people that are fairly new are gg'ing way too much. So did I but it got gradually less throughout the years as it's just usually misplaced.
After a loss and a spectacular game ofcouse it's quite okay to politely acknowledge this to your opponent but after a win? Never unless the opponent specifically does so.

whenever someone "handshake"s me it feels very nice and friendly. 

It's just people trying to be nice, but it soon gets formulaic and insincere so I've stopped bothering.

When a game is well played on both sides and nearing the end, I make a move which I think will be fatal and the opponent responds with an excellent counter move I did not foresee and I am honestly impressed, I often text "good move" . Response is normally "thanks'.  There is no shame in acknowledging talent.

quite right

About 15 moves into the game ask "so....have you ever played chess before?"

But you better win after that. 

I agree

This post made made me think of the Derek & Clive "hello" sketch (warning: strong language):

http://www.youtube.com/watch?v=GVMasdUbhfs