Yes, we usually use î at the beginning of the words but also at the end of a word. We never used it in the middle when it becomes â. We say câine, not cîine (dog), pâine not pîine (bread), and we also say a coborî not a coborâ, a hotorî not a hotărâ. To conclude, we use î at the beginning and at the ending of a word, and we use â in the middle of a word.
That's almost correct. There are some exceptions in words with prefixes:
neînțeles (ne+înțeles), neîndemânatic (ne+îndemânatic), neîncredere (ne+încredere), preîntâmpina (pre+întâmpina) etc.
There also exceptions in names. One somewhat famous example is the romanian tennis player Sorana Cîrstea.