It depends on the election. In primary elections, you can only vote for candidates from the party to which you belong. In general elections, you are free to vote for anyone you like, regardless of their party affiliation.

Yes. Iowa law allows for Election Day registration if you are able to provide proof of identification (i.e. valid photo ID) and proof of residency (i.e. utility bill with your name on it if your ID has an outdated address on it). If you are unable to provide one or both of these you are allowed to bring someone with you to attest that you have the right to vote.