Several questions!

Given the choice between monitoring through a home stereo and a good pair of headphones, I'll take the headphones. What you want is a flat response so that the speakers aren't coloring the sound. Another option is to get a pair of powered speakers designed for recording. Below is a link to a Sweetwater.com page listing active monitor speakers. If nothing else it will give you an idea about what is available and how much it costs.
 
http://www.sweetwater.com/c405--Active_Monitors
 

I may be in the minority, but I think the cost difference between a suitable pair of headphones for mixing and suitable pair of monitors gives a slight advantage to headphones.  The Audio_Technica ATH-M50 is $159 well spent: http://www.amazon.com/Audio-Technica-ATH-M50-Professional-Monitor-Headphones/dp/B000P62ND6/ref=sr_1_6?ie=UTF8&qid=1373424339&sr=8-6&keywords=audio-technica+ath-m50
 
They are no substitute for a good pair of monitors with a flat response and well treated room, but may7 give better results than your very likely colored home stereo loudspeakers.  There are definitely issues with mixing on headphones, but IMHO they are outweieghed by the fact that $159 will likely get you a set of weak, plastic monitors that are glorified computer speakers.
 
 

Answer:
)1 Ya you can do that, but EQ'ed ah, well, unless you have a 27 band equalizer (1/3 octave) and a test CD to measure frequencies with a sound level meter (Radio Shack use to sell one) and even with that it will still be subjective from the mic that picks up each frequency in the sound level meter, then the human ear chart for frequencies and loudness at which is played and anything else that is included before it suppose to sound good to you.
2) Ya, you can use headphones, I find ear buds to sound better than same priced headphones.
3) Your pick at the store, I like JBL, and since I do not have active monitors supposedly made for hearing (with correction built in like the two-way JBL monitors at $1559 for a pair)
http://www.sweetwater.com/store/detail/LSR4328Ppak/
 
I usually end up mucking it up, but listen to the stereo, headphones - and ear buds, and computer speakers and still muck it up, but I am learning and hanging around here found out some things I did not realize I had before. (like my RME unit has a Digicheck which shows the frequencies as they are playing coming out of the computer which I should have been using all along and probably should correct anything I have done, but there is always the next song I guess.
 
Live and Learn.
I figure none of it is perfect and since I done sound checking since vinyl days (sound level meter) that everyone hears different, every system is different, and all will sound different, but if it sounds almost alright through everything than I guess that is closer or as close as I may get even having the option of better equipment, unless someone wants to build me a $million dollar something or so studio like this one - Blackbird Studio C perhaps:
http://www.youtube.com/watch?v=exQRfHbSwn0
 
and that is only one of the 4 studios there.