Normally, you would be asking if Jack has any jackets at all (i.e. does
he have protection from the cold?). For singular countable nouns,
an article (
a/an/the jacket) is
always needed, unless it is replaced by a demonstrative (
this jacket) or a possessive (
your jacket). Generalization has nothing to do with it.
Jack, do you have a jacket at home?
Do you have (any/some) jackets is possible, but only if you are enquiring about the extent of Jack's wardrobe:
Do you have (any/some) sweaters? Do you have (any/some) anoraks? Notice that natural English usually includes the determiners
some or
any.
Re your second question, both are OK.
Who can be singular or plural, depending on context:
I love my wife, who is standing next to me.
I love my children, who are at day camp.
In your case, the plural
igloos recommends the plural verb:
Who live in igloos?
Who lives in an igloo?