Bullet selection is VERY important
The key to getting good performance from the .357 magnum in a rifle is bullet selection. Do a search, I have covered this before.
Because the rifle gives several hundred feet per second gain over the handgun, bullet performance is not the same. JHP bullets (and most JSP) are designed to function correctly at handgun velocities, and become rather "explosive" when pushed to rifle speeds. The only bullet type you can count on to act the same is the solid SWC, cast hard. These bullets are not designed to expand, and so at rifle speeds they retain their proper performance, increasing in penetration, if anything. However, the SWC is not a good bullet for small game, except with precise placement.
For example, I once saw someone put 3 rounds of Winchester 158gr Luballoy into a skunk, the first two without effect. The 3rd round clipped the spine, and took out the back legs. The first two shots were too far back, missing the heart/lungs, and gutshot, the skunk hunched at each shot, but kept on walking. A proper heart shot, or a head shot would have put down the pest cleanly, but the shooter didn't manage that. In fact, so distressed was this individual at the "failure" of the mighty .357 that he begged me to finish off the skunk with the .45 Colt handgun I was wearing. One round of 250gr SWC to the head put sent the skunk to whereever it is that shunks go for their afterlife.
Correct bullet selection AND proper placement is the key to making the .357 magnum a very useful and handy small game, pest, and short range deer rifle. Just remember that what you have is not a rifle, but a pistol caliber carbine. For small game and pests, the JHP/JSP bullets intended for the handgun work quite well, expanding violently, like varmint bullets. For small deer, and hogs, these same bullets, while working well enough from the handgun, often fail when fired from a rifle, blowing up and while creating a horrendous surface wound, they can lack the penetration to get to the boiler room and cleanly put the animal down. The hard cast lead SWC on the other hand, just drives right on, often completely through. Many people put too much emphasis on expanding bullets, believing that a bullet that doesn't expand is nearly worthless. Nothing could be further from the truth. While a properly behaving expanding bullet does give the best performance, combining expansion with penetration, an improperly behaving bullet (an expanding bullet when pushed too fast) often fails to deliver the desired results. Understand that an expanding bullet will only deliver proper controlled expansion in a certain range of velocities. Too slow, and it doesn't open up. Too fast and it opens up like a small grenade, usually before reaching the vital area of game animals. The 125gr JHP that has such a great reputation as a defense round from a handgun will kill like lightning small animals such as skunks, muskrat, even coyote with a proper hit, because when fired from a rifle, the increased velocity becomes a significant factor. Fire the same round at a deer, and chances are you won't be happy with the results. With the 125gr, only a neck or head shot should be taken, because the bullet will likely blow up on the shoulder bones and not penetrate to the heart/lungs. Standing broadside, where you can be sure of missing the shoulder bones ought to work as well, as even though the bullet will blow up on a rib, it will create a lot of fragments that will usually drop the deer. You might lose some meat that way, though.
The 158gr SWC will have the penetration for getting through the shoulder, or for raking angles, and will not blow up, if cast hard. Same bullet for hogs should do fine as well. You don't need expansion, you need penetration, and of course, proper placement. Note that the big 40+ caliber rifles used for hunting dangerous game in Africa rely heavily on "solids", for penetration, not on expanding bullets.
You are enamored of carrying "only one kind of ammo" for both rifle and pistol, and in a sense you do. But, if you want proper performance on a variety of game, while you will be carring only one caliber of ammo, you will want to have at least a couple of different loads. Solids for best performance on big game, JHP/JSP for pests, and I would recommend the lowly .38 Special LRN for a low (relativly for the rifle) speed load for edible small game. Hit a rabbit with a .357 JHP from a rifle and unless you make the head shot, you'll likely only have half a rabbit left! The .38 Spl will collect camp meat with a lot less destruction, even from the rifle.
Choose the righ ammo for your task, hit in the right place for the ammo used, and the .357 carbine won't let you down.
Sorry I can't give you any pointers on which factory ammo is best for what, as I handload extensively, and haven't bought or shot any factory ammo (besides .22LR) for years. I have pushed the 125gr JHP to 2200fps from my Marlin carbine (max loads), and they don't perform well at those speeds. Blow pieces of a varmint into the next county, but unless everything is perfect, only cripple a deer, and ruin a lot of meat in the process.
Tip: Some of the rifles don't feed the SWC bullet very well if you try to cycle the action too fast, or at an unusual angle. What happens is the round gets slightly misaligned, and hangs up on the edge of the chamber. You can feel this as soon as you start to close the lever. When (if) this happens, don't try to force the lever closed, you will only jam things up good. Pop the lever forward, and then close it. Usually it will close smoothly. What pushing the lever forward does is it takes the pressure from the bolt off the base of the round, allowing it to fall back into its correct position on the lifter again, and when it does, then it will chamber smoothly. Round nose profile bullets usually don't do this, and some rifles don't do it with SWCs, but if yours does, try it.
Also, be careful about overall cartridge length. I don't use the 180gr bullets in any of my .357s (rifle or handgun), but I did see one rifle jammed with some 210gr LRN handloads. The rifle had to be taken apart before the jam could be cleared. Rounds that are too long CAN go in the magazine, but will jam the gun tight when you try to feed them. If you want to shoot rounds like this, they must be single loaded directly into the chamber.
Enjoy your carbine, they are a great little gun. Just don't push it beyond what it is capable of, and you won't be let down.