YUP! I have a Henry .357 Big Boy. It's my favorite woods deer rifle! Sorry to take so long answering your questions, but I don't often monitor TFL.
On your specific questions...
Mine has a little side to side "play" in both the hammer and the trigger, though nowhere near as much as my old Winny 94 30-30 did.
And you are quite right, there is no "half-cock" safety. This is an honest lever gun. It has, or so I understand, a transfer bar so that it can not fire if dropped or something similar when the hammer is down. But other than that, the safety on these guns is the one between your ears.
Reliable? Well, I bought mine used over 2 years ago. Since I have bought it I have put over 2000 rounds of .357 through it (only 50 rounds of 38). It is the one gun that ALWAYS goes to the range with me.
I've never had a problem with it Never a FTF, FTE or anything else. At under 50 yards I always get a complete pass through on whitetail. Usually DRT. When they do run, the blood trail is so good a blind man could follow it at night.
I am so pleased with this gun that I also bought a Henry H009 30-30 and have a Henry H006M (.44 mag) on layaway.
When Henry comes out with a 45-70 at the end of this year (approx) I'm going to buy one of them too.
You have a great gun! Get out there and shoot it!