Not so sure about fiber. You likely can not do it the correct way which would involve cutting the fiber and putting end on it. It takes very special tools to cut fiber.
In general I don't know anything realistic that you would need fiber for in a house. Used to be you did it to get 10gbit but now it is easier to run that on copper. I guess if you wanted 100gbit but then you would need to run single mode fiber rather than multimode.
The best so called "future" proof is to run flexible conduit. This allows you to easily pull out the old cables and run something else in the future. You never really know I remember very well all the tech stuff saying SVHS video connections were the future and now days everything is HDMI or display port.
They make cables with coax and ethernet combined but it is likely going to be easier to just run 2 different cables at the same time. Not so sure how much actual need there is for coax. It is nice to use for moca when you don't have ethernet but I am unsure how much longer you will need coax. Most cable servers are now streaming and you can even stream inside your house between the cable box and the tv over wifi or ethernet.
In any case you cat5e is perfectly fine if you already have that. If it is a new install you might consider cat6a if you think you actually have a use for 10gbit any time soon. Do not use cat6 unless it happens to be cheaper than cat5e and cat6a is certified to go 10gbit where cat6 might work at shorter distances at 10gbit but it is not actually certified to do that.
Key to the install is use quality cable. You need pure copper wire (no CCA) with wire size 22-24 (none of that flat or thin wire). Also for in wall wires you want solid copper not stranded. Not sure exactly why but that is what the standard says.... mostly I suspect it is because wall keystone jacks are designed for solid core wire and not stranded wire. Do not buy from any vendor who does not clearly state these type of specs, vendors that don't tend to be selling fake cables.