They don't interact at all - because they're different abilities, even though they add to the same thing, you calculate them separately. So at 1, you get +1 from Stern Gaze, +0 from Favored Class. At 2, that becomes +1 and +1 (Stern Gaze is 1/2 level, it just has a minimum of 1, so at level 2 it stays at +1). At 3, it's still +1/+1, because you always round down (so 1.5 rounds down to 1 in both cases). At 4, they both jump to +2, and then every even level they'll each gain +1 again. So you'll never actually have an odd bonus except at level 1 - it'll go:
1: +1
2: +2
3: +2
4: +4
5: +4
6: +6
7: +6
8: +8
9: +8
10: +10
11: +10
12: +12
13: +12
14: +14
15: +14
16: +16
17: +16
18: +18
19: +18
20: +20
Third is where it gets tricky. Stern Gaze would put it at 1.5, and FC would also be 1.5, I see those numbers added together as 3, but my DM disagrees, saying that the .5 fractions wouldn't stack together to make a whole point.
Your GM is right. Those are separate abilities and as such are calculated independently. You do not get +1.5 and +1.5 adding for a +3 total. You get +1 and +1.