There isn't really a bad time to visit. Feb/March it will be warm, not too crowded, and the falls should be roaring - more water during the summer generally. More chance of rain as well. Between the falls and the chance of rain, choose clothes/shoes that will be comfortable when wet. As long as you avoid Easter you shouldn't have any crazy crowds. I recommend getting to the falls as early as you can in the morning - fewer people, more birds
Others have already covered Karadya and Surucua - I don't have experience with either so won't comment other than I believe Karadya is more affordable as a place to stay than Surucua? Guy Cox is a great guy and would be a good option if you're looking to go guided.
As far as where to stay in town - no particular recommendation but lots of options in Puerto Iguazu. If you find something affordable in the area mentioned above that is closer to the Hilton, east of Ruta 12, you'll have better birds around your hotel and if you leave your hotel on foot.
I agree with the recommendation for the Jardin de Picaflores, and the recommendation for the Macuco trail in PN Iguazu. As far as enjoying the falls themselves,there are three areas - a lower boardwalk, an upper boardwalk, and the Garganta del Diablo (accessed by train or walking). I like to do the lower two boardwalks in the morning, and the upper boardwalk is definitely better in terms of views. Then in the afternoon you can talk the train (or walk, good birding if you walk it) to the Garganta del Diablo, as you have better light most of the time this way. If you do two successive days in the park, the second day is 1/2 price, which is a nice way to take better advantage of the Macuco trail.
I am not certain the Brazilian side of the falls is really worth it, I greatly prefer the Argentine side, and the birding options are better on the Argentine side.
If you want to go completely independently/cheaper, you can visit the falls and the Macuco trail, bird around town a bit, but then you should think about renting a car and driving to Parque Provincial Urugua-i (free, camping only, bring a tent or sleep in your car), and also using your car to bird Ruta 101 through the park. Camping at Urugua-i also gives you an excellent chance of seeing Tapir at night. There is more great birding further away from Puerto Iguazu depending on how much time you have. The next "top" site with lots of different birds would be PP Cruce Caballero (a rental car might not get you all the way there but you can park at the entrance and walk the last km). Cruce Caballero is good for Vinaceous-breasted Parrot, Helmeted Woodpecker (with luck or help), Araucaria Tit-Spinetail, Antpittas, Owls, and lots more.
If you want more info on independently birding the area, feel free to ask, I spent almost 3 weeks camping and birding all over with a 4WD camper. Had a great time, saw all the specialties and tons more, I really love Misiones.
A side note, for the really motivated, the best forest in Misiones (and purportedly the largest intact tract of truly pristine old growth Atlantic Rainforest) is in a reserve called Papel Misionero, which is past Parque Provincial Caa-Yari some distance (South of San Pedro). Spectacular place and forest, but 4WD is mandatory and there are zero facilities there.
Suerte,
Josh
