Is it true that I can have some caffeine in pregnancy?

Yes. Limited intake of caffeine is safe: the equivalent of two cups of coffee a day. Not "venti", just a cup. Many women however lose their taste for caffeine or coffee during pregnancy. You may also find that you become more sensitive to caffeine, and this can lead to trouble sleeping. In those cases either avoid caffeine or limit intake to the morning.
Yes & no. Caffeine is a category c drug that is probably ok in moderation early in pregnancy but worrisome as you approach term. The newborns system cannot metabolize it easily (1 wk to eliminate 1/2) & they can get jittery, fussy. Pregnancy is a time to consider anything you eat/drink for the health promotion value.