Most of the house is wired with CAT5E, but I assume I can carry on using CAT 6 for the new cabling and this should work fine with the existing CAT 5E?
It will work fine, but the whole link is only as good as the weakest part of the Chain.
e.g. the following is Cat6 compliant
Device -> Cat 6 patch cable-> Cat6 patch panel-> Cat 6 Solid UTP Cable -> Cat6 euro module/face plate -> Cat 6 patch cable -> Device
the following wouldn't be
Device -> Cat 5e patch cable-> Cat6 patch panel-> Cat 6 Solid UTP Cable -> Cat6 euro module/face plate -> Cat 5e patch cable -> Device
Device -> Cat 6 patch cable-> Cat5e patch panel-> Cat 6 Solid UTP Cable -> Cat5e euro module/face plate -> Cat 6 patch cable -> Device
And whilst a lot of the time non-compliant cabling will work (especially as gigabit will work fine over almost anything), once you actually come to want 10Gb you may be in for a hard time and encounter random issues.
I've recently experienced this at work as we moved a few servers from 10Gb Fibre to 10Gb copper - we had Cat 6 patch panels, Cat 6 patch leads, but the cable between 2 patch panels was only 5e. For the most part it worked, but every couple of days we'd see a random server disconnect for a few seconds. The cabling between the 2 patch panels was only about 7 meters, so I ran some Cat6A patch cables direct between the affected servers and the switch (taking the same path as the Cat5e) and have had no further issues.
I put it down to it being a full bundle of 24 cat5e cables, and it being in a relatively electrically "noisy" environment, with a rack full of servers, hardwired UPS power cables running in nearby cable tray, and several electrical fuseboxes all within the Server room.
Note that it worked fine at 1Gb, and likely would work fine at 2.5Gb (which I may test when my 2.5Gb network cards arrive)