One way of looking at the bitwise & operator, is a method of setting bits off.
So we take the binary value 00000011, then take the result of that & your variable, all the bits where you used a 0 will be set off in the result, and the bits where you had a 1 will retain the original value.
So the result of (a & 3) will have a zero in every position except the last two bits. You test if that == 3 to confirm that both bits are a 1.
I don't think unsigned float would be as useful as unsigned integers, and there are not much hardware support for it.
You already have float, double and long double so we don't need yet another type. If C++ added a short float it would probably be the same size as float on most implementations because making it smaller would just slow things down because there is no hardware support for such 2 byte floats.
The internal representation of floating point numbers is optimised to match the hardware. The number will probably have separate sign, mantissa and exponent. Attempting a direct bitwise operation wouldn't make sense, as the three separate parts could be mixed up.
The point about using various bitwise operations on integer values is that the hardware and machine language is able to directly perform the operation, making it very fast and efficient. There are no such hardware equivalents for floating point values.