My experience with wads is limited to a few experiments with NECO wads under gas checked bullets. If I remember correctly, the wads didn't help things one iota, so I didn't pursue them further.
Gas checks make life far easier, even at lower velocities. Plain base rifle bullets are for someone who just likes to experiment, or someone who is determined to save money -- not a small consideration given the current price of gas checks.
So my suggestion is use a gas check, and save the plain bases for when you are bored and looking for a challenge.
BHN can go either way. Many standard cast rifle bullets are bore riding designs with undersize noses, not to mention undersize bodies. A soft bullet may obturate and make the bullet fit better, improving accuracy. Dave Scovill did a nice article on this subject once, testing different BHN's in a 30-30 with a bore riding nose. In his test, the softer bullets shot better because the undersize nose bumped up to fit the barrel (Dave never bothered to explain why the bullets didn't fit right to begin with).
Likewise with BPCR bullets, it is common for the nose to be undersize so it will chamber easily, and then the "bump" from the black powder obturates much of the nose to fit correctly. However, with smokeless powder and higher velocities, I've had better luck with hard bullets that fit right to begin with.
There is no simple answer to your question. Every gun is different. Mass produced guns are highly imperfect from the cast bullets point of view. Much of the cast bullet game consists of finding a bullet that compensates for the gun's imperfections.
For hunting, I use a hard (usually heat treated WW) bullet with a big meplat. This alloy will not mushroom, but it will usually "rivit" at rifle velocities, becoming close to a full wadcutter, and that is good enough with big bores.