Don't believe it costs anything, just fill out the form. Your realtor should have helped you through this process. I'd contact them and have them send you the form. Or you can probably google it and find it online.
Yep, you won't qualify unless you lived in the house on January 1st of the year you are claiming the exemption. You should be able to claim for next year, though.
You just take the forms to the Travis cty appraisal distract. Look up what docs you need. You need a car registration or something like that as well. Look at traviscad.org for what you need to do. It's free.