As far as I know the following is consistent with everything in this thread and the maps I've looked at:
The A38 has always gone to Liskeard and Bodmin.
No, not always. It originally started in Plymouth. Bodmin to Plymouth via the Torpoint Ferry, including the section between the ferry and city centre was the A389.
The A387 has always started at the junction south of Trerulefoot and west of Polbathic, and gone to Looe and Polperro.
Again, not always
. Originally it ran from Saltash to Trerulefoot. The road to Looe was the B3250 with the B3253 running Liskeard to Polperro via what is now the B3254/A387. Trerulefoot to Looe (but not Polperro) became an extension of the A387 by 1930 according to the historical mapping on SABRE.
...the A38 ran via the Torpoint ferry, while the A374 ran via the Saltash ferry to Trerulefoot. From St Budeaux, across the ferry and through Saltash, the A374 was the minor partner in a multiplex with the A388.
Originally the A388 started in Saltash with the A387 crossing the ferry at Saltash and running into Plymouth. Interestingly the A388 ran through the town centre and the A387 ran around the north of the town centre on the present day B3271. The road between Forder Valley and St Beaudeax - now the B3413, before that the A38 - started out as the B3246. It had changed to the A3078 by 1932. The whole route from Forder Valley to Trerulefoot via Saltash had become the A374 (originally used in East Devon) by 1946. The A389 via Torpoint to Bodmin had also changed to the A38 by this time.
The trunk road followed the A38 but was suspended through the county borough and restarted at Torpoint.
The Tamar road bridge opened in 1961 but neither the road numbers nor the trunk route were adjusted promptly.
The trunk route was routed via Saltash in 1963 (restarting there instead of Torpoint) but the road numbers were still not changed.
The road numbers were changed about 1967, and on the Cornwall side have not changed since. The A38 swapped with the A374 east of Trerulefoot.
Through Plymouth the A38 has followed three routes: first through the city centre, then along what is now the B3413, then the present route.