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HDPE is plenty tough, and should be more than adequate for your tiles.
A pocket hole jig would add about $20 to the cost, but is really simple to use, and let's all your connections be hidden so no spackling. The joint is also more reliable, because the screw is on an angle.
The challenge with cutting glass isn't a matter of how fancy the saw is, or how careful the operator is. It is a matter of having the right blade on the saw. There are 2 types - 1 for cutting tile, and a 2nd for cutting glass. The glass blade is significantly more expensive, so not a std part of even most expensive tile saws.
I made a toy box like this, but rather than buy the joiner, I just bought T&G flooring that's already designed to combine into a flat plane. A Liquidator store had the bamboo my wife like on clearance, so it was pretty inexpensive. Better yet, they had rough 3x3 pallet strap supports that were being discarded, that they allowed my to salvage.
Yes. Sinks usually have a template that shows the void required in the counter. Build a form the size/shape of the template, and pour around it.
You could also attach cove molding, or most any other trim you want/think would form well in the concrete. They make some here (US) in plastic that would likely release easier, but even if you only have wood trim, covering it with tape can create a reasonable surface for releasing the mold.
Very nice design and execution.If you're doing more cabinets, I would strongly encourage you to get a pocket hole jig, such as a Kreg, for hiding joining pieces of wood. I've done a few cabinets without one, so I was amazed and frustrated for not having picked one up sooner.