1965 Washington Quarter DDO?

Usually 1 photo doesn't do it justice when you are using a crappy phone camera. And if you actually look on the first photo at the B it clearly shows also on the 4th pic. All seem to exactly match the 001 on variety vistas.

Why would we need multiple photos of the reverse to identify a DDO?

And no, definitely not DDO-001. If you look at IGWT, you can see the hole in the O of GOD is the wrong shape, and the spacing of the bars of the E in WE are too wide to be DDO-001 (and none of the other letters show evidence of doubling either).

I'm really not the right person to answer your questions, as I know little about quarters, but I can add a few comments.

When you see the shape of the letters in the details, and they look like the DDO or DDR in the listings, some things come into play. First is to know what the letters are supposed to look like, which can be done by looking at a bunch of coins from that year on coinfacts and even ebay. Font types can vary quite a bit year to year, and some really do look thicker and similar to a doubled die depending on the year. The age of the die also plays a part in how they look, as they generally get thicker as the die ages, along with becoming familiar with Machine Doubling (MD).

But once you see a coin that looks promising, you hunt the online images of the doubled dies for a "PUP". That is a pick up point. I did that for 1965 quarters and see that the DDO-001 has a nice bit of notching on the lower left of ERTY letters. Your coin is in good shape, and if you don't see this clearly then it probably isn't DDO-001.

I also looked at the DDRs. I found the PUP for DDR-001 (DDR-002 it is not) to be the thickness of the crossbar in the R of AMERICA. It's quite thick to the point of strongly reducing the size of the hole in the top of the R. Your coin does not show this.

So while generally a great thing to recognize that doubled dies change the shape of the devices or add to them in some way, it gets down to the specifics for matching something to a known variety. See the odd coin out, and then work on finding a PUP that nails it down.

And while I do not see a doubled die on your coin, again I know little about quarters and could be missing something, that doesn't mean that there aren't still more great doubled dies yet to be discovered. In the end, if you are pretty sure it's some variety but can't figure it out, then you pay a couple dollars and send it to an attributor and they will do it for you. I end up doing this fairly often with the tough ones to figure out to know I have it done correctly and get the right label on the coin. And yes, sometimes I'm just wrong and it comes back as nothing. And even the experts can disagree. I have a coin featured in the current (September) Coin World magazine, a new Lincoln Cent doubled ear, while one attributor didn't even call a DDO.

Sorry I can't really help with your coin much, but seems like you are on the right track and wish you the best.