How are microchips designed?
Both. And there's quite frequently a team of hundreds involved; the latest Intel flagship processor will probably have had over a thousand people involved in design decisions somewhere (especially if you count technical input from the foundry, which is vital if you're using a new manufacturing process).

Generally the process involves:


high level architecture design, by humans
implementing the design in Verilog and/or VHDL, by humans
translating that into logic gates, by software
placing the gates and routing wires between them (software with constraints and guidance by humans)
analyse the result (software) and review key areas for improvement (by humans)
translate gates to images for photomasks (represented in the "GDS II" file format)
generate test data (software-assisted humans)
manufacturing and test process (mostly automated)


Edit: there are also lots of things which up the transistor count without having to design them all individually; SRAM, for example, is a large grid of repeating elements. The A7 will have a substantial fraction, maybe a majority, of its transistors devoted to L1 and L2 cache.