int(var) is an initialization and creates a new integer object. Its no cast.
Yes, it is called "function-style cast". It is to avoid like C-style casts because it has the same dangerous semantics (for example, it can accidentally cast away const).
The functionality of C-style and function-style casts is split into static_cast, const_cast and reinterpret_cast. Besides, C++ introduces dynamic_cast. You should use the specific operator for each conversion (mostly this will be static_cast). In general, a good style is to cast as few as possible, since many casts indicate a possible design flaw (wrong types chosen).
If you can cast, its probably faster.
Faster than what?