I've entered writing into contests and put excerpts of my novels places like Tumblr. If I wanted to seek publication for the pieces from which said excerpts are taken, could I still get published by a publisher (not self-publishing)?
For excerpts, it should be fine as long as you're not publishing too much:
If I publish an excerpt online, does that mean the whole work or part of the work is considered previously published?
Generally speaking, excerpts are okay to publish online, as long as they are on the short side (relative to the work in question).
With contests, it's a little trickier. When you submitted your writing, what rights (if any) did you give away? (You read the fine print to make sure you still have the right to publish it, right?) In any case, most publishers want the exclusive right to publish.