The new string is heavier than the original G, so will have more tension. This, combined with the fact that it's new while the others have settled, giving it a different tone, particularly from the bottom three, will make it sound different.
The tuning issue may well be because it's a new string. New strings take a while to settle in. If you do nothing but tune and play, in a few weeks, you'll find it's blended in, usually.
The idea of a set of strings is that they're matched to each other, all with approximately the same tension. That's now out of balance a little, and when you put the rest on, there will be more tension on the neck, given they make a heavier gauge set. Not a tremendous difference, but possibly enough to make the action higher. That, coupled with the tighter strings will make the guitar not so pleasant to play. Adjusting the trussrod, as user1079505 suggests, will pull the neck straighter, by tightening the trussrod by maybe 1/4 or 1/2 a turn. Not a recommended job for someone who's not experienced.
I have no idea why you have a heavier set of strings, but if you're happy with the feel of the old set, keep to that gauge, it'll save touching the trussrod.