Can I trade in my brand new ipad for an iPhone?

Yes, in principal you can trade-in a qualifying Apple device against other device. For dissimilar devices, you’ll need to do this in store - as the online trade-in system assumes trade for a similar device.

In store, qualifying devices can also be verified and immediately tested against acceptance criteria. You’ll receive the trade-in value by way of Apple Vouchers - which can be used against purchase of Apple devices of your choice.

Alternatively, you can use the mail-in service - although this can be more problematic. The condition of your devices and functioning, subsequent to their return, may become a point of disagreement. If you choose to follow this path, Apple will send shipping containers with which you can send your devices individually to their trade-in partner.  Upon acceptance, the assigned trade-in value will be sent to you by way of an Apple Gift Card.

More information can be found here:

Apple Trade In - Apple

How new?

If it's new enough, most retail stores will allow a full refund within a specified period, even if used. Apple Stores I believe are 14 days. Other than that, Apple's trade in credits are usually a pretty poor value as it will only be a fraction of the retail price as credit.