Does the word "beach" come from "bleach?"
Short answer: Most likely not.
Longer answer: A negative is usually more difficult to support than a positive. To say there is absolutely no connection between 'bleach' and 'beach' we'd have to know ... everything... and then scan 'everything' for support of the positive and have good reason to say that the support for each of them doesn't work. As that is not really possible we can only give likelihoods, things like in similar circumstances there was no connection. There are all sorts of single instances of phenomena in language that are not rule based, but rules (common patterns) are more common than the single exceptions (there's an argument that needs to be made for that but let's not go so quickly down that rabbit hole).
Here are two situations that are similar:
Phonetically, there is no regular alternation or sound change in English or in between word initial 'bl' and 'b'. In English there is no post-plosive liquid drop rule from proto-Germanic, it just doesn't happen. In other languages, Latin -> Italian has 'bl' -> 'bi', but the 'l' is not lost it is more changed to 'i'. (there is a bit of chicken-egg here because these rules must be found by having words of similar meaning sharing lots of similar phonology; but if you throw all the words in together before knowing the rule, a bunch of them would stick together and 'beach/bleach' would be an outlier.
Semantically, things are a lot more fluid. Word meanings can travel far. 'Black' and 'blue' are actually cognate with 'bleach' (flipping between light and dark colors). So it is not crazy to think that 'beach' is a 'bleached seashore purely by color association. It's just that the known etymologies don't show any evidence for their connection.
So to your 3 questions:
- No, there is no evidence beyond someone just saying so.
- We have no idea where that author got their idea (beyond the obvious similarity in spelling)
- There is no evidence
Looking at the first usages in the OED, we see that most of them mean pebbles, and the first one might easily mean pebbles.
1535: The smooth hard beach on the Sea~shoares burnes to a purer white.
1552: A Banke of baches throwen up by the Se.
1566: Wee haled your barke ouer a barre of beach or peeble stones.
1597 J. Gerard Herball ii. 249 Rowling pebble stones, which those that dwell neere the sea do call Bayche.
And Shakespeare seems to use it to mean the seashore, although twice it specifically means sections of the seashore that are covered by pebbles:
Coriolanus: Then let the pebbles on the hungry beach Fillip the stars;
Cymbeline ... can distinguish 'twixt The fiery orbs above and the twinn'd stones Upon the number'd beach?
So it seems to have originally meant beach pebbles, and later came to mean pebbled beaches, and from there it evolved to its current meaning of sandy or pebbled beaches.
So the real question is: did beach pebbles start getting called beaches because they were white, or because they came from streams, or for some other reason? One subquestion then is: are British beach pebbles white? Some of them are, and some of them aren't. I don't think we can conclude anything definitively one way or the other.