Is USA banning immigration from Islamic countries?
Not exactly banning from all Islamic countries, but suspending for a period from those countries where terrorist organisations have a significant presence.
President Trump plans to suspend issuing visas for people from Iran, Iraq, Libya, Somalia, Sudan, Syria or Yemen (where terrorist organizations have a significant presence) for at least 30 days.
The order says the government will give countries 60 days to start providing the information (required from country of origin?) or citizens from those countries will be barred from traveling to the United States.
Trump is expected to sign the order the week ending 27 Jan 2017.
Can Trump ban any individual country from Migrating to the US?
Yes, he can, according to the U.S. Code › Title 8 › Chapter 12 › Subchapter II › Part II › § 1182 > Part f
Suspension of entry or imposition of restrictions by President
Whenever the President finds that the entry of any aliens or of any class of aliens into the United States would be detrimental to the interests of the United States, he may by proclamation, and for such period as he shall deem necessary, suspend the entry of all aliens or any class of aliens as immigrants or nonimmigrants, or impose on the entry of aliens any restrictions he may deem to be appropriate. Whenever the Attorney General finds that a commercial airline has failed to comply with regulations of the Attorney General relating to requirements of airlines for the detection of fraudulent documents used by passengers traveling to the United States (including the training of personnel in such detection), the Attorney General may suspend the entry of some or all aliens transported to the United States by such airline.