1. The tap will leak because one or both the ceramic 1/4 turn cartridge (hot and / or cold) have partly failed and is / are letting water pass through when closed.
2. If the tap also leaks from the swivel point of the spout, the seals between spout and body have failed.
3. As ShaunCorbs says, multiple faults point to tap replacement.
4. As ShaunCorbs also says, it’s not a straightforward job and you might be well advised to get a plumber to do it. The point where the hot connects isn’t obvious and might require taking the back of the cupboard out. The new tap may well have flexi connections, which would require the existing rigid pipework to be extended. Minimum 1 hour, maximum probably 3 hours, so say 2 hours. Depending where you are in the country could be anything from £40 to £300 for labour (plus VAT, plus taps).
5. That’s the most pointless bit of earthing I’ve seen in a long time. It achieves nothing unless there is continuity through the flexible pipes. Personally I’d either leave it out if new taps have rigid tails (as the current one have). If extending pipework (and its copper), I’d bond on that new pipework.
6. A plumber can do that earthing.
7. Most unlikely the whole sink would have to come out. Its why plumbers have extra flexible bodies!
8. If you buy a decent make of taps (Bristan, Franke, Grohe, HandGrohe, various others) you should be able to get spares for a good few years afterwards. If you go for an unknown make, the taps may not be maintainable. You would have to pay for any “guarantee” replacements to be installed.