reinert,
You're welcome, I hope my experience is a help. My first deer rifle was a Model 600 in .308 Win (back in the early 70s) and I played with it extensively. I have since collected 600s in .222 Rem, .243 Win (Mohawk 600), 6mm Rem, .308 Win, and .350 Rem Mag.
The problem with the trigger is that it is essentially the same trigger used on the 700s, and the design can malfunction if dirt, debris, or thickened oil gums it up. IF this happens, the rifle can fire when the safety is released. And unmodified guns must have the safety off to remove a live round from the chamber.
Remington did a recall on the 600s, many years before the troubles with the 700 trigger came to light. I have never personally had any problems, but it is something you need to be aware of.
Other tips about the 600s, look at the plastic floorplate. It should be flush with the stock along the top edge, not bowed. If the floorplate is bowed away from the stock, it means that someone took the rifle apart, and didn't put it back together correctly.
The magazine box is a sheet metal "U" shape, open at the rear. To CORRECTLY reinstall the box, you must pinch it together at the back end, and fit it into the receiver opening. Then CAREFULLY put the action back into the stock. (what works easiest for me is to have the action upside down, and lower the stock on to it.)
IF you are clumsy, you will bump the mag box, and the sheet metal will pop "down" away from being fully seated in the receiver. This means that someone who does this, and doesn't realize it, can warp (bow) the plastic floorplate when tightening the action screws.
The floorplate can't force the mag box back into place (you have to pinch it together and reinsert it. Not doing that bows the floorplate. It doesn't change the functioning of the rifle, but it looks like crap, and for those who know, is a dead giveaway that someone didn't do a good job.
There are loads listed for the .222 using 60 & 70gr bullets. Velocities shown maxed about 2800fps from a 24", so 2500-2600 might be possible from the short carbine barrel. This will do the job on deer, but must be considered a precision instrument, meaning there is no excess of power. Personally I wouldn't choose a .222 for a big buck, but smaller deer, with shot placement SUITABLE to the bullet used, being critical, should give satisfactory results.
My book says Rem .222 is a 1-14" twist. Now, this probably won't do a good job stabilizing the 60gr+ bullets, BUT while it might group too poorly to be a good varmint load, it might be accurate enough for deer at 100yds, or less.
Only shooting will tell. Handloading with different powders & bullets might come up with an acceptably accurate combination, despite the twist being "too slow"
As an example, while I have never shot the heavy bullets from my .222, I have a Win 70 Varmint .22-250 with the same 1-14" twist. That rifle will do 3/4" groups with a 52gr HP, but only about 2.5" groups with the Sierra 63gr semi pointed spitzer. 2.5" sucks for a varmint rifle, but is good enough for deer at 100 or so. So don't automatically write off the heavy bullet in the .222, just remember its not what the rifle was built for.
You've got a fine little carbine, don't ask more than it was made to give, and you will be well served.