The DX200 Editor can't open sysexes. You need some other sequencer software. Which one do you use routinely?
DX200 Editor's host software XGWorks should do it. I'm checking this out - but at the first glance, here it's very difficult to start working with the empty project (and you need nothing more than an empty project, to insert sysex in it), it seems you need first record something over MIDI, at least one note. Awkward. I will let you know if I'll find something more.
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Checked. No, that would be living hell, not worth even trying. So you need other sequencer. I could tell you how to do this with Sonar or Nuendo.
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One more hint: if you tried it with MIDI-OX and that didn't work, check if the DX200's memory is protected. Shift+16, repeatedly 16 while holding Shift, until you see Prot. If it reads On, turn data knob to Off.

My usual DAW is Live, but I also have Reaper on Windows. The memory protection is turned off.

Sorry, then I can't help you. As far as I know, these DAWs don't support outgoing MIDI sysex messages.

By the way, don't you experience this infamous DX200's sickness, when you rotate the data knob clockwise, but the parameters are going backwards? Deoxit is not available where I live, so I bought some Taerosol, but still undecided if it is a good idea to apply it for cleaning/lubricating the contacts.

My encoder works fine, but the cutoff knob needs to be reseated or soldered. I've never used taerosol (Never heard of it) for cleaning encoders, but a lot of compressed air, or a small amount of alcohol has done well for me in the past. Put a little on it, and work it back and forth a lot.

Thank you for the hint, I was just afraid if the applying of the alcohol alone would be too much "aggressive" for the contacts. Taerosol (the product from Finland) just fixed it for me, at least for a while.