Are transcendental numbers computable?

Any algebraic number is computable; just employ approximation algorithms to obtain roots to any desired level of precision. Therefore, contrapositively, uncomputable numbers are transcendental.

An example of a number that is not computable is Chaitin's constant. Being able to compute it would yield a solution to the Halting Problem, which is undecidable, so we can't. Technically the constant depends on choice of programming language or universal Turing machine (I'm not entirely clear on the details there), so there are a whole suite of uncomputables in this way.

Yes, every incomputable number is transcendental, or, differently said, every algebraic number is computable. (Because it is possible to compute an arbitrary close rational approximation to every algebraic number).

As you noted, not every transcendental is incomputable. Consider for instance $0.101001000100001\dots$ (or $\pi$ or $e$).

A counting argument shows that nearly all numbers are incomputable: there are only countable many algorithms, so only countable many computable numbers, but there are uncountable many real (or complex) numbers.