too dry in Winter

Dear pals,

I need some advise about the following:

1. it is pretty cold these couple weeks, so we turn on the heat; however, I
find it is kind of dry for dd. Should I get some kind of humidifier? hot
one or cold one? what about vaporizers? air purifier?

2. dd's skin is kind of dry, especially on her face. Any moisturizer should
I put on her face? Does Vaseline good?

3. dd is 5 months old, and I haven't start feeding her any solid; however,
I found there are some rash on her chest (not itchy). Does it caused by
dryness? What kind of cream should I use to get rid of those rash?

Please help. Thanks in advance for any suggestion or comment.

Jen