Been some time since I played Minecraft, but I'll give these Qs a go
Can I run Minecraft on a windows laptop?
>> Yes. If you have a laptop currently, you can check a site like https://www.systemrequirementslab.com/cyri
to see how well it will run
Do I need HDMI connection to the TV?
>> If we're still using the laptop, no. You can play it on the laptop screen just fine (but if your TV has a HDMI port then it's an option). If you're taking the console approach as below, then your TV will need a HDMI port.
Can I run the latest Minecraft on an old xbox? I see lots of xbox one for sale on donedeal.
>> Minecraft is available on both Xbox 360 (i.e. the old Xbox) and Xbox One (if a second hand Xbox One counts as an old one, then yes
The PC version tends to get the latest and greatest updates and new features (I believe, but I could be wrong on this), but the console versions have also been supported. That is about to change as the Xbox 360 version will not be updated
after the Update Aquatic. It looks as though Xbox One players will continue to get updates.