If I don't know who's calling I always say prosím. If I know who's calling and it's my friend I ussually say some informal greeting like čau and if it's someone else I just say prosím.
I always answer calls saying my name. It is because I also like much more when my calls are answered with name then with just simple "Prosím!" or "Čau" (this one I hate the most" :x ). I feel not at easy when I should answer the call with my name at crowded place but I still do so.
I guess it really depends on the person. I never answer phone saying my name. I don't know why. I just don't do it.
I've also heard "Slyším," although this only from a 70-year-old grandma. The joke I'd always play on her when she answered the phone that way, was to reply, "Jsem rád, že ještě slyšíš."
Taking the cellular implementation of push-to-talk into consideration, the proper answer should be now: "Tady orel, tady orel. Jak mne slyšíte? Přepínám." :wink:
Thanks for the responses! I had thought there might be a single word that everyone uses (such as PRRRRRRONTO! in Italian). But I guess it's more of an individual thing in Czech