My DD was very upset and concerned that she could not have a chocolate finger thsi morning. I was dismissive. Hey ho.
The two things are NOT the same at all & you know it.
I have been this child, granted a couple of years older, only my mum didn't wait till my dad left & she had an affair instead. One that I caught her at. Dad left, new man moved in that night. I didn't hate him, but I resented the hell out of them both because I wasn't given the space & time to come to terms with my dad leaving, my mother having an affair & my having found her at it.
As an adult, when you refuse to put your child first in such circumstances, you cannot be too shocked if said child starts to feel pushed out, inadequate, upset.
Child need time to process things, just because the op wasn't mentally in her awful marriage for some time, it doesn't mean that she can just override the needs & feelings of her child.
The partner here is already sulking & stopped the majority of communication with op. The op admitted that she spends all day, every day on skype (which conveniently went down to school hours on her being pulled up on it) & the man seems happy to encroach upon said daughters space by telling the op that she shouldn't change anything. What kind of adult dismisses a child like that? One that doesn't give a damn!
And it makes me wonder if he would feel her feelings & boundaries are so unimportant if he did eventually move in with them.
He doesn't want a 13yr old dictating to his girlfriend, yet HE is happy to do it? 
the OP does need to address her DD's real upset over the breakup of her family. Which is what everyone is saying.
That doesn't mean not dating, it doesn't mean keeping her BF a secret Again, that is what we are saying.
and it doesn't mean taking seriously every complaint of her DD about her BF. I would say it is pretty fucking important to take a teens concerns over her bf seriously, actually. She doesn't have to act on the all, but she DOES need to listen & she does need to help her dd process those concerns & take them away or fix them for her.
Adults decide, children need to accept. No no no & a thousand times NO!!! What if this guy is trying to work his way into ops house because he has a thing for 13yr old girls? When he starts to disregard her boundaries by walking in on her in the bathroom, or in her bedroom, should HE decide that that is what HE wants & the child accept it? You are sending out a message that adults are always in the right & to contest an adult, or their actions is wrong.
I am not saying that this is what is happening, but you cannot dismiss a child's fears & feelings because her mothers right to date is more important than her right to feel safe & comfortable in her own home. Which she doesn't, when he is around.