Which sounds better according to British English?
Does climbing down make good sense and is it idiomatic?
Disclaimer: I'm not a native English speaker, but until some of them arrive here's what I've got:
This Google Ngram shows that all 4 options are fine, with similar frequencies of usage (BrE 2012):
where climbing is slightly preferred. It probably isn't a BrE/AmE difference, because (according to Google) climbing is a slightly preferred option in AmE as well - see these Ngrams for climbing up vs. going up and climbing down vs. going down.
I'm guessing that you are talking about actual physical ladder (not social or career one). LDOCE seems to prefer climbing:
lad‧der [countable]
1 a piece of equipment used for climbing up to or down from high places.
In conclusion: while climbing up or down is used slightly more often, going up or down is also fine. To your 2nd question: yes, climbing down is definitely idiomatic.