Assuming there is no screw or onsert on the top the handle should just pull off (it may need a bit of leverage underneath the cap against the spout) chances are the tap has had it. fit a tap reviver kit that replaces the top and working parts reusing the base fitted to the basin cheap as chips and easy DIY!
Yes 1/2" for basins and sinks, 3 /4" for baths. They work well except in some cases of bath mixers they can be a bit troublesome to get the depth correct but basins and sinks no problem.
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