Well, you can include a photo in a form, but it’s not really the same as having a paper form with a logo on it.
Yes, you can email a completed form. The next update to Tap Forms will also let you include photos and audio file attachments in those emails.
You can search an individual form for any text or numeric value or you can search the entire database. I don’t have the ability to search just a single field. It will search all fields of a form in one go. I hope to add a more advanced search function in a future update.