CBDunkerson
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"You need both hands free to climb, but you may cling to a wall with one hand while you cast a spell or take some other action that requires only one hand."
So yes to attacks, but no to TWF or 2HW.
The climb speed is important to retain your Dex to AC.
What you want for combat is something like Slippers of Spider Climbing... so you can have both hands free even while moving.