just throwing my opinion out here:
In regards to him going on tour in Europe, like someone else mentioned perhaps he was only going for part of it. Also, if he knew in advance that it was being cancelled I doubt it would be hard for him to resume filming for his show, it's not like it is some overly high quality/expensive production that they can't just pick it up and make content when able to. Also him going to a concert while she was sick with an illness that made it so she could not sing is not that strange, she was clearly not incredibly ill but was advised not to sing, his life shouldn't stop and I doubt she would think it should just because she is sick.
Also, before this interview/talk show I am sure there was some pre-planning in regards to what to ask him. I don't think he deflected or changed the subject to avoid confirming they broke up. This was probably discussed before hand since rumors have been circulating for a while that they've broken up and they probably asked in advance if it was something to avoid talking about it and they likely would have avoided it if he said so - which would have been much more solid confirmation that they broke up rather than speculating on a question that he answered "YES" to. If he deflected it it was probably to avoid talking about their relationship because the man can't even post an instagram photo without getting 700 "where's Lana" comments and the man has decades of life behind him and she and him are probably both aware that their relationship would be a huge focal point despite them both having their own separate lives of accomplishments.
We don't really know that they haven't seen each other in over a month, she barely posts anything and he seems to be traveling a lot of the time so who knows. I wouldn't be surprised if they meet up in New York when they can, or wherever. It's not that they haven't seen each other in over a month - we just haven't seen them together in over a month and I don't think that speaks any kind of volume. She is getting a lot of negativity for dating a cop, and he's also been affected by it. I wouldn't be surprised and would also support them keeping their relationship more secretive for those reasons.