Coax - Yes. It's a wideband connection suitable for Freeview, Freesat and VM router (if you use triple-shielded HD100 coax). I'd run two or three to cover most bases.
HDMI - No. The standards are still changing, and although sending bog standard 1080p HD over 10-15 mtrs is easy, you won't get much more than 5m on copper cable at 4K and even less when 8K hits. Use Cat6 cable (×2 +1 spare) and then add devices called Baluns on either end to convert 1080p/4K plus a control signal to in to something that can travel 20-30m.
Network cable - Yes, but bare-end copper Ethernet not the aluminium-based extension leads with the plugs fitted. This is because of speed and reliability. 1x live and 1x spare will cover it.
Some may advise just using wireless as new technologies seem to offer Gigabit-busting data speeds. Wireless should be included in your house plan, but it can't always be relied on; there are too many ifs, buts and maybes that will drag the speed down. A bit of copper cable running at Gigabit speeds will be 25 times faster than you need for streaming BT's Ultra Hi Def 4K sports which is currently the most data-hungry streaming service.
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