Yup, that's the pressed checkering as used on the later 340s. I'm not exactly sure when the checkering style was switched so I'll venture a guess that your rifle was made sometime between the late 50s (starting around the timeframe of the B or C improvements) and 1965 (since it has no serial number).
As for doing a trigger job on the 340s. You can clean up the feel bu polishing the sear surfaces but there isn't much you can do to lighten them up. I've done a couple that wound up in the 5-6lb range but they do feel lighter with the rough machine marks stoned off. Like any other trigger job, it takes finesse and attention to not change any of the angles on the mating surfaces otherwise it can go from clean to unsafe in a hurry..
I never did try to mess with the initial takeup and overtravel though so I can't help you with that part of a 340 trigger job.
It sounds like you might have some rust or grit inside the trigger groupl that is causing the intermittent erratic trigger. I would start with a good solid cleaning and check for burrs on the parts. Also check the sear spring to make sure it's strong enough to fully return the sear when the action is cocked. A past owner might've fiddled with the sear and or trigger spring to try to lighten things up. Both are fairly simple legged coils springs so it is possible to tweak them by bending.
The trigger parts are 29-35 on the parts list.