The Henry does cycle plastic dummy rounds so there should not be an issue.
Probably not. But its not a guaranteed certainty, as "plastic dummy rounds" are not dummy rounds they are snap caps. True dummy rounds, used for function testing replicate live ammo in ever aspect except for firing. They are the same dimensions as live ammo and the same weight. And normally of the same materials (brass, lead, etc)
Plastic or aluminum snap caps can, and usually are functional through the action, and are a fair approximation of live ammo, BUT not an exact one, and I have personally seen a case where the snap cap worked and a loaded round did not. It is a VERY rare thing but it is not impossible, and I mention it just so you will be aware that, unlikely though it is, its possible your snap cap may give you a "false positive".
My experience with .38/.357 lever guns is with the Marlin, so only generally applies to the Henry, but here it is, for what it is worth.
Length matters, but not too short as much as too long. Bullets seated too long will jam tryng to go from the magazine to the chamber. (if you are at or below industry spec for length, it doesn't happen)
Bullet shape matters more than you might think. Specifically the SWC shape. My Marlins feed anything with a rounded nose shape butter smooth, no matter how you work the lever. RN lead, or JHP/JSP bullets, everything with a rounded sloping nose (even with a flat tip) works great.
SWC on the other hand like to hang up. Feeds fine, but ONLY if the lever is worked a certain way, at a certain speed. Too slow, or too fast, and the round "bounces" on the lifter and the sharp edge of the SWC bullet likes to catch on the edge of the chamber. When this happens, the solution is simple, just back off the pressure on the round (move the lever forward, slightly) then the round will fall back into its proper position and feed smoothly when you close the lever.
This MIGHT not be an issue with you Henry, but if it is, its not a deal breaker.
Most lever just work best when the action is worked "smartly", meaning not to slow or slamming hard fast. But every one is a little different, and your rifle will show you what it likes, and doesn't as you shoot it.
Stick with standard .38Spl loads, the rifle barrel will boost the speed up to or past +P levels anyway. And DO CRIMP. Tube magazines tend to push the bullets into the cases unless crimped.
Good Luck, have fun, let us know what questions you have.