Is the language $\{a^nb^{2^n}|n\geq 0\}$ Turing recognizable/decidable? I have proven that it is not regular and context free. But is it Turing decidable/recognizable? I'm thinking yes, as I am able to write a Java program for it, and by the Church-Turing thesis the power of the Turing Machine and a Java program is equivalent (or am I wrong here?).
You are right about the fact that being able to conceive a Java program for doing the job, then there must exist a Turing machine which will also recognize that language. However, you don't need the Church-Turing thesis to confirm your statement. You need something weaker, namely the fact that both Java and TM are acceptable programming systems and hence they are isomorphic (ie. they compute the same set of partial functions) by Roger's theorem.

The main point is that Roger's theorem is a theorem and Church-Turing thesis is not proven.