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It's on the other side of london from you but I'd just go and hang out there to get a feel for it. The town centre can be a bit bleak but it's not unrelentingly so. I don't mean to be down about it - I really like it - I just mean it's not 'nice' nice. I like that it has a little theatre and a library, and half-pedestrianised town centre (market, town square, 'arboretum'), well-used park right by the town centre (Abbey gardens). There's a bunch of new stuff going up by the river Roding - some of it looks pretty crap but the stuff from about 10 years ago around the old granary is settling in nicely and last time I was there had some creative studios, restaurant, cafe etc. That's all between the river Roding and the train tracks. Most of the housing is on the other side and there aren't that many crossing points. I don't know how snarled up they get in rush hour. There are suburban residential areas where it can feel a bit endless and far from anything except a massive park.
The bit to south and east of your outline is a bit of a no man's land. Do not recommend. Barking Riverside / Creekmouth is new-build hemmed in by light industrial. Lots of families but transport access is pretty crap and you'd feel very isolated. They were meant to extend the DLR but it got scrapped. Also bear in mind about Dagenham or south of the A13 is it's usually downwind of the Beckton sewage works. Visit the Sunday market at least once though.
Don't know if you're old enough to remember the BNP in Barking breaking through, but yeah, there's that. Recent history. I think there's been a conscious push back since then, but more recently it was also strongly brexit (the whole borough, don't know the local breakdown).
@Sumo - I guess Barking is too expensive? I like Barking. It has a proper little town centre, night bus, fast train and slow underground into London. Becontree has a nice feel and some nice bits but no centre, it's spread out in a suburban kind of way with shops kind of scattered about. Dagenham has nice bits and bleak bits. I only know Romford from cycling through a few times, the centre feels pretty bleak and very car-dominated, there are quite a lot of 'nice' well kept suburban streets but I'm guessing they're expensive.