Well, you'll probably get away with it, but you might not get the full speed out of it, it depends on the length of the run, short lengths should be ok.

Not that it really matters because our internet isn't very fast.

Ours isn't too bad (40-50 Mbps FTTC) but we're in the process of getting full fibre FTTP installed, they've run it along the street and into the building, but it's not active as yet so we can't get it connected up, hopefully it will be complete in a week or so, then we should be able to get 900 Mbps 

Ours is fiber to the NID and it does about 500mb/s. If we change service providers for one that is faster, we would probably have to change some infrastructure. But it looks like even with your new service, you would be within the limit of CAT5e cable.

Probably just about, but I tried to future-proof the installation as much as possible, so we went for Cat6.

Yeah I'd go with at least Cat 6 or even Cat 7 for a modern install. Most of the wiring I have is Cat 5E since it's older. Still works great for 1 gigabit and probably can push it a bit higher with new NICs and switches but since my internet is only ~300 megabit, it works. Higher local network speeds are quite nice when you need to transfer between computers or between computers and a NAS though.

Currently the only internet service in my area is via coax or POTS lines to the demarc but they're installing fibre to the premises now. The rough in for fibre to the houses was completed a few weeks ago, might be up and running by the end of the summer.

Most of the desktops and stuff like game consoles are wired with the except of a couple in awkward locations for running ethernet to. Laptops I sometimes plug in but I mostly rely on wifi. That said one of my laptops has a terrible wifi card on it and I'm stuck using it since Lenovo decided to use a wifi whitelist for that model laptop. Otherwise I'd probably would have upgraded the wifi card on it.

I just don't trust the security of a wireless connection. Plus whatever you use it's always going to be slower than a wired link and we store a lot of our data on NAS drives. As I've said, we're hopefully going to get a 900 Mbps connection in the near future, so it will make a difference there.

We mainly use the laptops in bed, so I installed sockets either side, see here. When we're away we have little choice but to use wifi, but I can't say we've had any problems, apart from the painfully slow connections you always get from hotel wifi 

Oh, I strongly recommend using a VPN, especially on public systems. You don't know who's out there...

Oh yes, Cat5e is way out of date, you really need Cat6, I think Cat7 is around now as well, but we use 6 here. Got a good deal on a full drum of low-smoke cable 

Unless you are planning to run a 10Gb ethernet over 10's of meters, cat5e Is fine.
For VPN, if you are using https and other enceypted standards, you won't have any issues.

As I said, I got a good deal on a full drum of low smoke Cat6 cable, all that was wrong with it was the box was a bit tatty..! Cost me £75 instead of anything up to twice that or more..!