Let's Burn Up a Character

I'd assume tinkering tools = mending tools, so that makes sense to me. I'm not convinced that any hammer or mallet included in such a set would be worth jack squat in a violent confrontation, though. If you want a weapon, it'll have to be bought separately.
Not having a weapon is perfectly valid. Burning Wheel has combat mechanics, but it is not tied to them the way D&D is.

EDIT: Resource points only exist in the context of character creation, and any that remain unspent are lost. Spare cash costs 6 resource points, and that's from the Man gear list, not the Dwarf gear list.