If the ball is located between yard markers, which yard line is displayed on the scoreboard and is the running back credited for a yard gain or not?
There's no rule governing this, but typical practice is to use the last integer yardline that was crossed. So if you're on your own half of the field and you're at the 42.5 yards, then you're said to be at the 42 yard line; once you cross the 50 and yard numbers decrease as you advance, then if you're at 42.5 yards you've reached the 43-yard line.
This also applies to specifying what yard line needs to be reached for a first down, which has the effect that there are "long" and "short" yards. So, for example, the actual distance on 3rd-and-1 can be very nearly two yards, if e.g. the line to gain is at 42.9 (so 42) and you're currently at 41.1 (so 41). Conversely, you can have two yards to go whose actual distance is closer to 1 yard, if you're at 41.9 (so 41) and you need to reach 43.1 (which is stated as 43).
Whatever the person running the scoreboard wants. The scoreboard is not official. The operator can choose whatever. For instance if it is 4th and 3 inches and the first down is right at the 43, most scorekeepers will chose the 42 to signify it is not a first down, although they could do whatever they want.