Any idea on what the Japanese dockyard numbers and production would be compared to the US?
Sorry I don't know what's in the game, but we will find out more about Japan's capabilities in the next DD.
@Axe99 would be the best person to ask regarding actual capabilities as he's compiled a spread sheet listing all ships produced during the war. I think he found that US tonnage constructed was greater than the rest of the world combined and Germany actually produced more than Japan. I'm guessing that US production capability would be 5 times that of the Japanese.
It's a tiny bit rough, as I haven't filtered out the classes that were being built across the game start date, but assuming these aren't going to swing the analysis too far one way or another. For ship types included in HoI, for classes whose first vessel was commissioned in 1937 or later (IRL, even destroyers often took longer than 12 months to build, so a 1937 launch date means classes the majority of whose production occurs during the time frame), Japan produced 0.85 million tons of ships, compared with the US' 3.76. On CVs alone, Japan produced 334,200 tons to the US' 994,400 (which is actually pretty impressive, if you consider the differences in industrial capacity).
However, and this is a huge however, HoI4 doesn't include all types of warship (HoI4's ship types cover less than 50 per cent of the tonnage of fighting ships used during the conflict), and the big exclusions are amphibious warfare ships (amphibious warfare ships combined come to 6.9 million tons of shipping), of which the US built the vast majority. If you compare overall 'fighting' ship production (so amphibious warfare ships such as attack transports, LSTs and LCVPs, as well as CVEs, escorts (of which Japan actually built 188,000 odd tons of, even if she was not that great at using them) and mine warfare ships), Japan's total production increases to 1.7 million tons, while the US' increases to 12.9 million, or 7.3 times Japan's production. Even the UK produced just over twice as many tons of fighting ship as Japan. Once CVEs are included in carrier capacity, Japan's 494,200 tons produced doesn't look quite as scary compared with the US' 2,347,700 (the US lend-leased a few hundred thousands tons of these to the UK under lend-lease, but if we're just looking at production capacity, they should come under the US' side of the ledger).
Japan doesn't start that badly though - counting ship classes where the first vessel was completed in 1936 or earlier, Japan has 1.2 million tons of warship to the US' 1.9 million (split across two oceans). There is a window at the start where Japan can compete with the USN, but once those US dockyards get moving, things get ugly quickly. If ship times are what we've seen so far, then it's likely that things will get ugly around a year sooner for Japan than it would have historically, which is what a few of us are concerned about.
Edit: PS - very good guess there
@jamesd 
.