What is the process of (digital)IC design?
I'd say that proper digital IC design is not done bottom-up but top-down !

You start with a specification of the function which you need. You get this from someone else or write your own.

Then you translate that into Verilog or VHDL and simulate until it does what you think it should do.

Then you generate the netlist from the Verilog or VHDL.

Very likely you will simulate that netlist to verify that it works as expected.

From the netlist a layout is synthesized using the IC manufacturer's libraries. These libraries contain the layouts of all the gates needed (Inverters, AND, OR gates etc) and it contains information on the delay of these gates.

So normally digital designers are not bothered by the actual transistors as the gates are blocks which are ready to use.

The synthesized layout is send to the manufacturer for further processing and mask-making.

The wafers are then processed using these masks.

What I wrote might be wrong in some respects as I am an Analog IC designer, not a digital one. When I need some digital I just draw a schematic with gates directly. This is only suitable for simple functions with only a few gates.
I do sometimes layout my own transistor level gates though.