I personally don't see a problem with the ammo in the picture; assuming they are in fact factory loads! I shoot the same factory loads in this 1901 Single Action. Just make sure your Single Action is in fact in perfect mechanical and integral condition.
However, shooting these 100+ year old, very valuable toys, is always risky to both the investment and your eyes/health; be safe and take all precautions!!!!!
I am not up on this aspect anywhere near well enough, but being as .38-40 is both a Rifle Cartridge and a Revolver Cartridge, in the old days, when it was a Black Powder Cartridge, it did not make any difference which it was fired from, other than one got a higher FPS out of the Rifle of course.
Now though, things may be different, in so far as Smokeless will allow there to be dedicated Rifle ( 'Rifle Only' ) Loads, and, Revolver Loads.
If there are off the shelf 'Rifle Only' Loads, they may or may not be clearly labeled as such, and, one would not want to use them in an old Revolver.
.32-20 is definitely this way, and I just do not know whether .38-40 is this way, so...one would want to find out about that I think.
Have stuck this photo up in the past but I shot off this box of .38-40 factory loads in the 1905 4 3/4 Colt Single Action Army the day the photo was taken to have fun and provide additional empty cases for handloading projects.
I wouldn't shoot the Remingtons on the right, but just because they look older & the older ammo is very interesting to me.
I'm not sure how much older they are, but if they were mine, I'd keep them.
Enjoy that nice Colt, I'm probably not the only one who would like a range report when you get time.
Just one question. The more modern box of cartridges on the left.
I know it says 180 grn Soft Point, but I have a feeling that they're jacketed Soft Points and not totally all lead. Do these have a copper jacket? If so, I think I would leave those out of that old Colt. The other ones on the other hand are in fact Soft Points and are all lead so those won't harm your gun at all.
The jacketed rounds will in fact produce higher pressures than the plain old lead bullets will.
If you check the cartridge belt in the picture I posted, those are the same Remington 180 Grain Soft Points (jacketed) that you have. I've never had a problem with Jacketed 180's through any of my old Colts or Winchesters in my 40+ years of shooting them, but you need to do what you're comfortable with. I believe most of the Cowboy loads are all lead and loaded to lower pressures than standard ammunition, so like I say, go with that if you feel better about it. I guess if you were shooting 100+ rounds of jacketed ammo a week through the old Colt you'd see some wear at some point.