.38's won't gain much speed in the rifle, if any. Fun to shoot though.
.357's ought to get a good 300fps increase, more or less, from the rifle as compared to a typical revolver. Some loads more, some less.
Hunting - it's the bullet not the gun. A good WFN (Wide Flat Nose - see www.beartoothbullets.com
for pictures of various bullet styles) will make a pretty good wound channel out of a .357 revolver. I've shot a few small deer, trust me, you wouldn't want to get hit with one. What the rifle does is add range - at the point where your rifle bullet loses 300fps downrange, then it's equivalent to a bullet leaving the muzzle of a .357 revolver. Make sense?
Don't use JHPs for hunting from the rifle, they'll most likely blow to pieces. Use hard cast, or perhaps the Win - Nosler Partition handgun bullet. Speer Gold Dots are supposed to hold together well, but I haven't shot any game with them.
Gas-check bullets help with accuracy (as compared to plain based lead bullets) when velocities go up, but that's true with any gun/bullet, not just Marlins and not just .357s.
Tinkering around with cast bullets in rifles isn't for the impatient. Either you enjoy it and the chase is it's own reward, or stick with jacketed for simplicity of load development.