Transcendental numbers are non-algebraic, that can't ever be the solution to an algebraic equation. But how can they be identified otherwise? Is there a method of determining whether a given irrational number is transcendental or not?

No. All algebraic numbers are either irrational or rational. But some transcendental numbers (Pi) are not even algebraic. Of course all transcendental numbers are irrational, because all rational numbers are algebraic. Being algebraic is a proper subset of algorithmic derived numbers ... interestingly the cardinal number of algebraic numbers is the same as the cardinal number of rational numbers (dealing with infinity allows certain paradoxes as legitimate ... and this is why finitists reject infinity in maths). So an algorithm can be devised to derive all rational numbers (they are all algebraic). But no algorithm can be devised to derive all irrational numbers. Such an algorithm can be devised for all algebraic numbers (so all irrational numbers that are also algebraic). But there is a set of non-algebraic numbers, that are irrational ... that can be computed by an algorithm(Pi or e). There are plenty of well defined irrational numbers, that have not been proven to be transcendental (non-algebraic).

Proof falls under the same bug-bear as any other algorithm. There are constants, which haven't even been proven to be rational vs irrational, let alone algebraic vs transcendental.

https://en.wikipedia.org/wiki/Transcendental_numberSimple examples are not proven as to what class they are (Pi plus e).

One could try to escape this by allowing proofs that are not covered by an algorithm ... a proof with an infinite number of steps ... but that would be pulling a rabbit out of your hat. These are called oracles in Turing machine theory ... a way of formalizing cheating. Though in fact, there is no way to cheat. Basically it amounts to the proctor (oracle) handing you the answer to the question.

The question of what is a true random number vs a pseudo-random number, is critical in cryptology. A truly random number could be a normal transcendental for example, because by definition there would be no possible converging algorithm to calculate it. There may be degrees of true randomness, as the normal transcendentals aren't the only non-algorithmic transcendentals, but others nearly as bad. This comes up in computational complexity theory (which again plays into the subject of cryptology). I studied all this 2 summers ago, when studying cryptology.