An off peak day return from Newcastle to Alnmouth is £13.40. It has restriction code B3 which I think is just off peak.
At that price you're referring to the fare that's routed Any Permitted. There is also a slightly cheaper TPE only fare, which is highly inadvisable to buy due to the tiny saving compared to the loss of flexibility, particularly in view of TPE's continuing unreliability and inadequate ticket acceptance arrangements.
The restriction code on the Any Permitted fare is B3, which is a very common one that only just trains timed to depart between 04:30-08:59.
I presume I can break journey at Morpeth (only want to do this on outward)?
Off-Peak tickets permit break of journey unless the restriction code says otherwise. Code B3 doesn't have any break of journey restrictions, so break of journey is permitted on this fare (on both portions).
In the afternoon there is a gap in trains back and planners seem to send you via Berwick. Is that right?
It appears that some, but not all, retailers are willing to offer you an itinerary via Berwick during that gap (between the 15:26 and 18:00 direct services). I'm not sure how they are deeming this permitted, but you are always entitled to travel in accordance with the itinerary that a retailer offers you when you buy your ticket. That doesn't necessarily guarantee a hassle-free journey though, it has to be said.
If so, is the ticket valid on all TOCs - TPE, Cross Country, LNER, Lumo?
It's routed Any Permitted, so yes, it's valid on all operators along all permitted routes