Learn Italian: Sono stato talmente maldestro che ho fatto un pasticcio.

Yes, absolutely. Sono stato così maldestro che... means exactly the same thing. Both talmente and così are used with adjectives to introduce a consecutive clause, meaning "so [adjective] that".

If a woman were speaking, would 'talmente' change to agree with her gender?

No, talmente is an adverb meaning "to such an extent", so it is invariable and never changes its ending. However, the past participle and the adjective do need to agree. A woman would say: Sono stata talmente maldestra che...

Why does the 'che' clause use the indicative ('ho fatto') rather than the subjunctive?

Clauses of result (consecutive clauses) using talmente ... che describe an actual, factual outcome. Because making a mess is presented as a real event that actually happened—not a hypothetical result, a goal, or a negated outcome—you use the normal indicative mood.

Is 'fare un pasticcio' a set phrase?

Yes, it is a very common everyday expression. Literally, un pasticcio is a baked pie or a mixed-up dish, but figuratively it means a complicated situation, a mess, or a botched job. Fare un pasticcio is the standard colloquial way to say you messed something up.