There are no objective 'in universe' answers to the question you've posed. We can make some educated guesses and on-the-back-of-a-napkin calculations if you'd like:
How long does it take a ship to produce another ship equivalent to itself?
If we assume that the constructed fleet seen at the end of the book "Excession" is roughly equivalent to a "replicating event" then the answer is that the single largest ship-type seen in the Culture fleet (a GSV) can probably be constructed in as little as 20-30 years, assuming adequate materials were to hand. This tallies nicely with the events of the Idiran war, in which we see the Culture create a brand new class of vessel, the GOU and ROU (roughly half the size of a GSV) which then takes approximately 15-20 years to go from design to production.
The largest GSVs seem to agglomerate in size as they age, starting at around 25KM and growing to as much as 200KM at the upper end of their lifecycle.
Does productive power scale linearly with ship size?
Yes, absolutely. The principle, well established in the Contact novels, as well as in Bank's own "Notes on the culture" is that a General Systems Vehicle or a completed Orbital is an holographic representation of the Culture, capable of reproducing any object or structure found elsewhere in the Culture up to and including a viable Culture population
Smaller ships (Offensive Units and the like) seem to be restricted to producing their own drones, munitions and small "lifeboat" style units controlled by AI cores. The production of mind-controlled craft is left to larger System- and Contact- Class vessels as well as Rings.
What are the bottlenecks for exponential growth?
Culturally, the Culture spurns exponential growth. They view this as crass and akin to the actions of an homogenising swarm. Per Banks' own notes;
"why? What is the point? To put it in what we might still regard as
frivolous terms but which the Culture would have the wisdom to take
perfectly seriously, where is the fun in that?
Interest - the delight in experience, in understanding - comes from
the unknown; understanding is a process as well as a state, denoting
the shift from the unknown to the known, from the random to the
ordered... a universe where everything is already understood perfectly
and where uniformity has replaced diversity, would, I'd contend, be
anathema to any self-respecting AI."
Which part of a ship is the hardest part to build?
Arguably the most valuable part is the Mind. We see a factory tasked with the production of ships in Consider Phlebas. They prioritise the Mind above all else when the facility is attacked and there is some suggestion that the production of a Mind is considered the highest faculty of Culture technology.
Are ships constructed by nano-machines or effectors?
It is only within the most recent Culture novels that there's been a mention of increasing use of eDust and nano-machines in the Culture's culture. The majority of production in previous novels is either managed by effectoring pieces into their correct configuration or, as we see in Player of Games, allowing humans to have a small (and largely trivial) role in physically manhandling the parts into place.