No, the 300 ultra mag cartidge is not interchangeable with the 300 win mag.
It might not be chambered for 38 special.
Should be marked on the barrel or may have come with the directions or package what rifle do you have?
NEVER USE AMMUNTION IN A GUN THAT IT IS NOT CHAMBERED FOR!!!!!
It would almost certainly be chambered in .32 S&W Long.
NO NO NO! Only shoot what the rifle is chambered for and so marked.
If the weapon is chambered in 357 Magnum, you can fire the round in it.
Common sources include the loading manuals from Speer, Lyman, Hornaday Loading manuals for companies like Speer and Hornady are only going to give you load data for THEIR bullets--not Remington bullets. I haven't checked Lyman yet but the load data published by powder manufacturers are typically going to list loads using bullets manufactured by the major bullet companies. I am looking for a source that publishes loads using Remington bullets myself. No, you can't simply substitute a load using a bullet of the same weight as the Remington bullet you want to use. Bullet composition and design impacts CUP's.
The extractors break, particularly if shooting the "accelerator" bullets.
With a heart, spine, or lung shot, a .35 Remington will kill an elk. Load it with 275 grain bullets. With leverevolution powder from Hornady, a .35 Remington is much more adequate than you need it for elk.
The lips of the magazine are designed so that a cartridge will not be able to leave the magazine until the cartridge is pushed forward when being chambered.
I think your Remington Model 760 (there were other models also called Gamemaster, but not in 30-06) has a magazine capacity of 4 cartridges.
They made them in four calibers. 25 Remington, 30 Remington, 32 Remington & 35 Remington. The 25 & 32 are obsolete, the 30 was still available in the 80's & the 35 is still available at your larger sporting goods stores. There are dozens of custom reloaders that will still supply ammo for any of these calibers.