Are there general rules for Base + Metal Compound reactions like there are for Acids?
On the same level of generalization, the rule is "no reaction", but this rule has quite a few exceptions (probably more than the rules you quoted above).

Water-soluble bases would react with amphoteric metals to produce hydrogen and the corresponding salt:
$$2\ce{Al + 2KOH + 6H2O \to 2KAl(OH)4 + 3H2}$$