Should my partner still talk to their ex? They are still in contact everyday (all though I have been assured this has toned down), they never open messages from their ex while I'm around.

Honestly, I do not believe it is very healthy for a partner to keep contact with an ex while in a relationship. It can create trust issues and isn't very helpful in moving forward. If they have moved on, they don't find value to contact them everyday and hide their messages from you. I believe it would help to carefully address the situation with your partner. Having a relationship means getting feedback from each partner, so if you are uncomfortable it should be mentioned. You are not jealous for making the decision to confront. This situation isn't typical and can lead to bad things down the road, such as distrust.