The safest way to charge your car is to install a charging point at home rather than using public charging points. This means that you can all but guarantee that your personal charging point won’t suddenly become damaged or unsafe to use in the way that you might find a public charging point damaged when you pull up to use it.
That said, you should endeavour to have the charging point checked every year or so to ensure it’s still working at the standard it needs to and hasn’t been inadvertently damaged in the intervening time.
If you need to use a public charging point, it’s unlikely that you’ll get an electric shock even if it’s not working. It will probably just not connect to your cable and not charge particularly quickly or effectively. However, the public charging infrastructure of the UK is a slight concern for drivers who have to make longer journeys, and it could be a potential problem for you if you’re expecting a certain charger to be working and it’s not. You may then have to find a backup, but whether you have the battery life to do so could be another matter.
You should also be aware of the cable you use to connect your vehicle to a charging point - not only is it something that needs to be in good working order, but it can also be a trip hazard. If the cable crosses a path or somewhere that people are going to step over it to get past, you should use a cable protector to reduce the risk of anyone tripping over it.
Additionally, you should never daisy-chain your cable by plugging it into a multi-socket extension lead that then plugs into another multi-socket extension lead. This increases the risk of the connection overheating and starting a fire - you shouldn’t do it with any electrical appliance, but especially not an EV.
As good practice, you should make sure you get a new cable if you begin to notice wear and tear on the covering of the one you have - anything with an exposed wire is very hazardous in terms of potential shocks and electrocution.
Yes - you shouldn’t have any safety issues if you’re trying to charge an electric car in the rain. The cars and charging equipment have been designed with this in mind. If you’re charging at a public charging point, it won’t necessarily be under cover, but the equipment has been designed to be waterproof so there shouldn’t be any potential problems for you to worry about.