The non-magical weapons and armour available to characters depends upon the army that the character fights for. Every army book has an equipment list printed right before the discussion of the units and characters for the army. In that equipment list, any item that does not have a '*' by it is available to the characters of that army.
Champions of units must be armed and armoured the same as rank and file members of the unit (With a few exceptions such as Chaos or Undead Champions). Character points must be expended to equip unit champions.
Characters may attack the Fleshounds with non-magical weapons (temporarily sheathing that magic Frostblade to use a regular axe, etc.). All special characters come with 1 non-magical hand weapon. A character may change weapons in a combat freely.
Source: Warhammer Design Team / Warhammer Magic, p. 32