You might get gigabit over cat5 but it could be a bit hit and miss.
I have HDMI extenders working over about 20m of cat6. Tried with cat 5 and was iffy on a 10m cable.
Depends on the length of cat5, but we routinely used cat5e to feed gigabit to the desk in corporate buildings.
Gigabit works fine on 5e in pretty much any domestic environment. Better to buy lengths to fit, with connectors pre applied (with boots to protect the tag) as the usual point of failure is the crimp point. I'm rubbish at crimping, but I do it twice a year, some colleagues can do 50 a day!
If they genuinely have just Cat5 cable then it's only good for 100Mbps
Originally 1Gbps Ethernet was going to need Cat6 over two pairs. However they then realized that using all four pairs and a slightly enhanced cable aka Cat5e (the 'e' being short for enhanced) and you could run 1Gbps Ethernet over it. That's what won in the market place.
10Gbps Ethernet really needs augmented Cat6 cable aka Cat6a, but it is rated for use on Cat6 as long as you don't have large bundles of cables and only up to about ~50m. For most domestic properties that's fine.
There is now a NBaseT standard that allows 2,5Gbps over Cat5e at the full 100m and 5Gbps at shorter distances. With Cat6 you can get the 5Gbps over the full 100m distance.
Cat7 is not certified by the IEEE for use on any
Ethernet standard. In fact it needs special plugs and sockets to be used otherwise you will actually make things worse. Given that there is zero equipment available for purchase that has such sockets it is only use by the the gullible who think because it has a higher number it's better. A fool and their money are easily parted.
Cat8 is similar to Cat7 and needs special plugs and sockets not to make the situation worse. However this time there is an IEEE Ethernet standard for it's use, but it's only for use in a data centre, only for short distances and there are no products on the market for it use. I doubt it will ever get much use as evern 10Gbps over twisted pair is rarer than hen's teeth in a data centre. It's all direct attach twinaxe cables or actual fibre optic, as 10Gbps SFP's are as cheap as chips these, and both use less power.
Most domestic properties Cat6 is the way to go for future proofing. If you live in a mansion then Cat6a makes sense. My pro tip is to spend a little extra and get purple jacketed LSZH stuff. Makes it much easier to identify what the cables are for with a quick visual inspection in the future when you have your head under the floorboards.