Answer to Question 1
The gases in the Universe right after the big bang would have been hot and should have emitted strong blackbody radiation. The large redshift of the big bang material relative to Earth would lengthen the wavelengths of that radiation into the microwave part of the spectrum. It is accepted that the cosmic microwave background radiation is the radiation from the big bang.
Answer to Question 2
According to the inflationary universe hypothesis, the entire observable Universe was only about 1059 light-second across at an age of about 1036 second, there would have been enough time to equalize temperatures before inflation. This explains why the background radiation now has the same temperature in all directions.
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