Male-Male competition limited to breeding season?
It's strongest in the breeding season but male-male competition exists outside of the breeding season too, e.g. European Robins only breed in the summer, yet maintain territories year-round. It's pretty circumstantial, however sexual competition will only exist in the breeding season. 

To address the Red Deer example, the deer only compete with antlers in the breeding season, but outside of the breeding season they aggregate in same-sex groups too, so most of the competition males will experience is male-male. 

Normal behaviour will depend again on the species. Most birds I can think of have discrete breeding seasons and behaviour outside of this is rarely dependent on sex, generally including foraging etc. However I can think of many examples to this, such as Capercaillie, where the hens sometimes forage together. I suppose that 'normal' behaviour could be defined as any behaviour that is not part of the organism's motivation to reproduce.