And yes AFAIK what you're doing sounds fine, I used to do the same. If you mix warm milk with chilled you'd raise the temp of the chilled milk which would be less than ideal, but if they're both chilled before you combine them to store - no problem.
If you plan to use the EBM within a week or so, though, no need to freeze at all.
It's been year since I did this so want to check I'm not going to create a health hazard. If I express a couple of times a day into different, sterilised bottles, putting each into the fridge after and putting the pump in the fridge in between, and then mix and pour the chilled milk into a freezer bag and freeze it that evening that's ok, right? Then wash and sterilise everything for the next day.