I've always used a bowl inside the sink - it's the way I was taught. As has been said, it uses less water. It's also useful if you have pans or mugs that have been filled with water and left to soak until washing up. You can just pour the dirty water into the sink itself, leaving the washing up water cleaner. Many houses in the UK only have one kitchen sink, so it can't be poured elsewhere.
You are also right in your comment about breakages. Glassware and china is less likely to be chipped on a plastic bowl than on an enamel or metal sink, which used to be more common.

In the UK, it's common to have fine, knitted or woven dishcloths, like these from Lakeland; http://www.lakeland.co.uk/F/keyword/dishcloths The Lancashire cloths are the most common type. Some poeple use them for washing dishes and they are also used as general kitchen cloths, for wiping up and cleaning.

Thhis is a guess, but it may be that tea towel is used to distinguish a towel which is used for drying teatime things, (and by extension, the things for other meals, but tea towel is alliterative), as opposed to a towel used for drying the person. Most people prefer to use differnt towels for the different purposes.