It looks like Sedum reflexum. That isn't a bulb but a succulent. I don't know how much you cut them back but they will probably survive the pruning. Usually I just trim off the dead flower stalks in the spring and remove any that are growing where I don't want them. Marcia

It kind of reminds me of a petunia. I recall one of my reseeded petunias having stranger foliage than the parent plant, looking somewhat like yours...

Congrats on your new house! You may want to look at pictures on Houzz for some inspiration. Then you'll be able to decide how you want to use the space and whether it's something you can execute or if you need to hire a landscape architect for help with the design. When it comes time to choose plants, information about your climate will be important. You may also want to get a soil test done.

Oh, if you're in oz then it could be a few more things. What about desfontainia spinosa? Not that it's native there but that it would be Hardy. There are also some holly-leaf banksia but this doesn't look like them to me.