Generally speaking, try to develop your pieces unless you have to deal with some tactics or threats immediately. I have run across this a few times myself, and I also enjoy playing unorthodox chess. Consider an orthodox opening: the French. After 1..e6 white says "ok, no center? Well then 2. d4". Same story for flank openings in my opinion. Knights out, bishops active, king castled...with that setup you probably are ready to move forward and attack. Don't play automatic, mechanical chess though! Good luck.
Playing *against* alternative&unorthodox openings

The mistake to make when facing flank openings/unorthodox openings, is to think that because they violate cherished principles they somehow should lose immediately.
White has the first move... and the worst opening (f3 IMO) doesn't do anything too much worse than lose White's opening tempo... IOWs the worst white opening, is hardly worse than starting the game with Black!
In general a flank opening uses the hypermodern idea: "give em enough rope and they'll hang themselves." In this case, the rope is 'center' -- You are allowed to build up a center which the flank opening opponent hopes to use as a target.
Openings like a4 and h4 obviously make an early rook lift possible.
But the main thing to remember when you face these openings IMO is: you still have to play chess! Flank openings _aren't_ horrible refutable always-losing openings, not even against when it's titled player vs. titled player. At the ordinary player level where inaccuracies abound... these openings are quite playable!
For ideas of how to play, check out some games on the game explorer here. And go ahead and take the center. Your opponent thinks he's a hypermodern master? Ok. Prove it.

These are good replies. The basic opening principles apply:
1) Develop your pieces, getting as much space for them as you can. Some pieces have "favorite" spots in the opening, such as c3/c6 and f3/f6 for the knights, bishops out to b4/c5, etc. Don't bring the queen out too early, as she becomes an object of attack. While you are moving her again and again out of danger, your opponent is developing his or her pieces.
2) Fight for control of the center. Occupy the center with pawns and back them up with your developed pieces.
3) Castle. Protect your king and develop your rook.
Again these are guiding principles, not laws. Watch what your opponent is doing and "play chess" accordingly.

Playing slight devil's advocate here. I usually say develop in a straight foward efficient manner etc. But almost all unorthodox openings have method behind their madness. They are not neutral, instead have lopsided strengths and weaknesses.
You have to perceive the thought behind them and then play the developing moves that respond to them, sort of a rock, paper, scissors game when you get to see what your opponent has first.
In your 1st example:
Hello,
Recently I've stumbled upon some correspondence games wherein my opponent has used "unorthodox" or very uncommon openings. I can't say the exact opening I'm up against - for that would be cheating - but what do you do when you see openings like: 1. e4 e5 2. Nh3 or playing against strange openings like 1. a4? I never have an idea what principles to start upon to dictate the way I play against my opponent's who use these openings and take full advantage of their weaknesses.
Any insight would be appreciated. Please do not use just the examples above that I gave, for that would not answer my question.
photray94