O2 offers a basic PAYG Sim which doesn't require a credit check. But it no longer offers 'traditional' PAYG (where a customer tops up using credit and uses that for calls, texts and data). Instead, it offers data bundles, which start from £10/month.
The cheapest O2 Sim costs £6/month and it currently offers unlimited minutes & texts and 4GB/month data over a 12-month contract. See cheap O2 Sims.
Yes, you can keep your old mobile phone number. If you're moving from another network, you'll need to text 'PAC' for free to 65075 to request your PAC (porting authorisation code). You will need to provide this to O2 which should move your number over within one working day.
If you are already an O2 customer, you'll need to fill in a 'keep my number' form on its website.
O2 allows you to use your allowance in the EU without extra charges. However, if your plan comes with more than 25GB of data, you'll only be able to use up to 25GB of data within the EU, which is more than enough for most. For more info, see Cheap Roaming.
Yes, O2 offers 5G on all its Sims. However, be aware you'll need a 5G-compatible phone too.