Yes on both!! You will just need to use a couple of tricks and/or Add-Ons.
To make it so they can only vote once, in the settings (click on the cog wheel thing), require them to be logged in with their Google Accounts, then check "Limit to 1 response."
What if they don't have a Google accounts? That is going to be a likely scenario (PSH!). In that case, be sure to NOT check this option, otherwise it will block those non-Google participants from accessing your Form. With this scenario, there is really no way to guarantee that they will only vote once; you could add a field to collect email addresses, and then just eliminate the second entry. If anonymity is a concern, don't worry because it won't be anonymous either way. Just assure your participants that their votes will be kept confidential (and then be sure to keep it that way!).
To activate the submission window so that your Form will close after a certain date/time/number of responses, you'll want to use the "Form Limiter" add-on:
Once you enable the Add-on by clicking on the link above, you'll activate it like this: