Skype is officially banned, but as long as you've got it installed before you come out here it will work. Don't know where you're living, but the UAE has high levels of fiber penetration and in the main cities you can easily get 32 Mbps connection if you want it.

Don't know about your other questions, sorry. But there are only two telecoms companies (Du and Etisalat) so your options are limited, unless you can find some sort of international calling card that gives cheaper rates.

Personally, for simplicity's sake I'd leave a message on your US line answerphone advising people to call you on your new UAE number....