Surface code ideas make contact with several different areas of physics and coding theory. Toric code introductions from error-correction perspective by J. Haah and condensed-matter perspective by M. Levin and C. Nayak are listed in the EC Zoo entry: https://errorcorrectionzoo.org/c/surface.
Is my little summary of techniques more or less complete (and correct).
What is the most seriously considered technique to perform logical gates for the surface code?
Lattice surgery is the current best technique.
What is a good pedagogic reference to learn it?
For the basics I'd recommend reading "Surface code quantum computing by ...
A partial answer to Q1:
Braiding and lattice surgery are 2 different things. Just keep reading about it and notice that braiding is moving of holes on the surface, and lattice surgery is stabilizing together 2 different logical qubits.
Answer to Q3:
After spending a lot of time finding the best place to learn the basics of surface codes, I came to the ...
A really well-made resource explaining Grovers algorithm is 'How the quantum search algorithm works' on Quantum Country by Andy Matuschak and Michael Nielsen.
I think this fits your question nicely, as it is a fundamental algorithm, the explanation does not assume a lot of prior knowledge and the explanation focuses on a conceptual understanding.
The most useful use-case I found for a big fault-tolerant quantum computer, besides Shor, is a simulation of molecules, which are too hard to simulate on classical computers + demand fault-tolerant computers.
Read this article
Generally speaking, you can see the following demand (depending on the accuracy you want):
The approximation that they are doing ...
Copying over from "Are circuits with more than 1000 gates common?". Note that a Toffoli gate is roughly as expensive as 2T gates or 4T gates, depending on your architecture.
According to Table III of https://arxiv.org/abs/2011.03494 , quantum chemistry algorithms looking at properties of the FeMoCo molecule use half a billion Toffoli gates.
Topological quantum memory by Eric Dennis, Alexei Kitaev, Andrew Landahl and John Preskill is a very nice introduction to the surface code. It develops intuitive description of the code in topological language from the low level combinatorial properties of the lattice focusing on the planar surface code with hole-free encoding.
It is notable for its breadth ...