Based on what you are telling me, I would say that you are looking for a detailed, airy set of open cans. Something with a big soundstage and good instrument separation will be a good compliment to the M40x, imo. I still don't know the prices of stuff there, but I am going to list some around that price range which would probably fit the bill.
You've already heard/read plenty about the DT880 and DT990 if you read all the stuff in that link that I posted, so you should have a good idea of how those sound. If any of those sound interesting to you, any of them would likely fit the bill.
Then there is the Q701, which I have and like plenty well. The sound signature is a bit different/less reference when compared to the DT880. What the Q701 does have in spades though is detail, speed, and instrument separation. These are all really important to me because of my love of fast and complex metal, so it might be less important to you than getting your hands on some neutral sounding cans. The soundstage does lend itself to classical/orchestral music very well though, so if you have an interest there, these might be worth some serious consideration.
The SRH1440 is a bit out of your price range, I believe, however they have tons of detail. I am not too familiar with these, but they are worth a look. The whole Shure line is pretty interesting.
Audio Technica has some decent open headphones such as the AD700(x) and the like. Again, I am not too familiar with them, so you should look into some reviews on your own. You might like what you find, you might now.
As far as dacs and amps go, the O2 and ODAC are pretty hyped, I'm sure that you've heard of them in the past. They are a pretty solid option. I personally have the Fiio E17 and find it to be pretty decent overall, though I would like to replace it with something a little better. I personally like Schiit's products, though I don't know how available they are over there with them being an American company and all. You should look around for that. Open back headphones tend to require a decent amp more than closed back. The Q701 and DT880 both have a reputation for needing amping to reach their full potential. Taht said, as I am not familiar with the rest mentioned, they might also require a lot of amping. You can get by with most without an amp though, if you need more time to save up. I see no problem in doing that. The headphones are by far the most important part of an audio line up, so start there.
I urge you to do some reading by yourself of reviews of stuff that is available to you and in your price range. What people say first hand is often the best indicator of how things will sound. I personally like reviews on head-fi.org, but there are plenty of other places to find reviews (I like to avoid professional review sites as they could possibly be enticed to say one thing or another, whereas user reviews on a forum seem to be more honest and straight forward).
This is a great utility for getting a general comparison of some headphones. I generally like to use the Sennheiser hd800 as a reference as I like their sound signature. It isn't the end all be all by any stretch of the imagination, but it does offer some nice learning tools.