L.H.
Hey A., this all sounds normal to me (I have 4 year and 18 mo. daughters). Here are the best things I have found (from friends, pediatrician, etc.).
1. Tantrums are not acceptable. My doc said to place her in a timeout as soon as the tantrum ends so she learns that is not acceptable behavior. In a public place, you pull aside out of the way and speak a firm NO ______ (yelling, screaming, whatever is the offending behavior). He said not to take the child home if that is what they want, because it reinforces the yelling and tantrums as an effective device for getting their (wrong) needs met. That idea is nice, but you have to feel it out, I've found. By the way, we use the pack 'n play for the timeout spot (otherwise, the 18 mo. old would be everywhere). It is empty and away from the "action" and she knows it's not fun time. One minute (at least) for each year of life (so 1 min. for 20 months, but 2 mins when she turns two).
2. When you ask your child a question, "no" is a natural response of many kids this age for two reasons: One is that they are testing us as parents and looking for the limits of their independence (the limits that they desperately crave and need set for them to feel secure). The other is that NO is something they can say as an immediate response. If you give them some time and talk (I know that's hard at such a young age) they can "change" their mind but really they're just processing what's happening and "No" is the placeholder while they figure out what they really think. This does not help the repeating "no"s of a tantrum, but it does help the everyday communication (did for my first). I often asked a question that was reasonable for her to answer, got a "no", waited, and then got a "yes". Then I would repeat, "So you want/choose..." and when that was confirmed we went from there.
It's tricky dealing with this during pregnancy when all your senses are heightened as well (I remember), but you'll make it!!! I do agree that ignoring won't make this behavior go away. Keep looking for solutions that work for the parenting you and your husband choose and that work for your daughter. And remember, give something a try for at least a week or two - most of these patterns take some time to be effective. You'll see how she responds and can go from there.