I read that a stereo amp is better, since if the stereo channels have opposite phases then the mono amp might cancel or distort the signal.
Stereo channels do not have opposite phases per se, but you're close. On a keyboard that plays stereo samples, there can be phase cancellations at some frequencies on these sounds when the left and right channel are collapsed to mono. AFAIK, this is not an issue for keyboards that don't play samples, or where the keyboard (or particular patch) only uses mono samples. It comes up most often with piano sounds, which are almost always sampled in stereo. So one thing that makes your question hard to answer is that you haven't mentioned what keyboard(s) and/or sound(s) you use. There are lots of instruments and sounds where there are no phasing issues at all. In fact, once you factor out acoustic piano sounds, probably most keyboards/sounds hardly if ever exhibit phasing issues when collapsed to mono.) Also see:
http://forums.musicplayer.com/ubbthreads.php/topics/2250006/Mono_Piano_Discussion
The other reason some people prefer stereo amps over mono is just that it adds general spaciousness to the sound. Even when playing mono sounds (which is most of what's in most keyboards), there is often stereo spaciouslness added through the use of effects (reverb, stereo chorus, etc.). But the amount of this effect you will get out of a stereo amp is typically constrained by the fact that the left and right speakers are so close together, so two speakers you can put some distance between would generally be preferable for this. The other variable is whether this amp is being used only for your own monitoring (i.e. the audience is mostly hearing you from the main PA system and not from your own amp), vs. scenarios where your amp is providing the sound for the audience as well.
1) if a mono output amp has stereo input does that ensure the stereo sound will be handled well?
No. On a mono amp, there is no difference between using two mono channels vs. using a single "stereo input" channel, as far as I've seen. The benefit of the stereo input is primarily that you can make adjustments like volume with a single knob, instead of one knob for left and one knob for right.
2) what's better in case of a mono output amp (with stereo input) - send a mono sound from the keyboard to the amp (connect only the left/mono output) or send stereo and let the amp combine the channels?
I'd say the first option is better, because there's at least
some chance that the keyboard is doing something to make sure the mono sound is as good as it can be (I know at least some Nords have a control for this). However, more often, either approach only gives you a simple summing, so there should be no difference.
Here's an important missing piece in the discussion so far: As you alluded to, most keyboards have a left/mono output (though occasionally it may be right/mono), which as I said, typically just sums the signal, which can produce phasing issues in some cases. The trick to avoid this when playing mono is to use the side that is NOT labeled mono. That is, if a board has a left/mono out, it can be preferable to use just the RIGHT output... which will give you only half the signal (just the right side, no summing with the left signal) but will eliminate any possible phase issues. There are trade-offs, but it can sometimes be the best sounding option. On a piano sound, using just the right side will typically emphasize the higher notes of the piano over the lower notes. But that may give you a better piano sound overall than combining the two sides to mono. If you prefer, you can put a "dummy" plug into the right output, and then the left output will give you just the left side of your signal (instead of a mono summed version), in which case your lower notes will be emhpasized over your higher ones, but that's rarely the better choice. This dummy plug approach may be more useful if you run into a keyboard where it is the right output jack that is used for mono summing, where otherwise the non-summed output would normally give you a piano sound that emphasizes the lower notes over the higher ones.
I'm not really worried about having a stereo effect, more that the mono amp won't ruin the signal due to phase issues etc.
If you're not worried about any stereo effect per se but only about phasing issues, the info above should help you out. Don't get too freaked about the "problems" of running mono... tons of people do it. Tons of house PAs are mono (or run that way), too. As long as you find a piano sound you can live with in mono, you should be fine. Many boards have "mono piano" patches for this purpose. Also, some boards make it easy to integrate external sounds, so if you're happy with your setup except for your piano sound, you could possibly grab a piano sound from, say, an iPhone/iPad.
the amps are usually quite heavy anyway, plus the price of two good amps is going to be a bit high.
So I think I would only want to buy only one.
What's your budget?
FWIW, a good quality stereo amp (e.g. Motion Sound KP-408S) is not so different in price or total weight than, say, a pair of EV ZXa1 plus a small mixer. The Motion Sound is more convenient in that it moves/sets up as one self-contained piece. The pair of EVs gives you more flexibility in placing the left/right sides, and also allows you to spread out your purchase... you could start with one speaker (playing mono) and add the second later to get the stereo if you decide you'd really prefer it. You'd also have the option then of bringing either one or both of the speakers to a given gig (maybe there are situations where you'd prefer to travel lighter), plus if you have two separate speakers there, you have backup if one fails.
