What kind of effect did this Pokemon have during its time in the 5th Gen NU Metagame?
This Pokemon basically causes every single team to have basically a counter to this or at least multiple checks. Sawk is also considered the main Fighting-type on NU. Fighting-type resistances are common as hell because of Sawk's influence because without one, every mon essentially gets 2HKOd or OHKOd (most offensive mons are OHKOd). It's been in NU forever and started to shine after the first start of NU as people realized its brute power. Without an answer to Sawk, you're basically asking yourself to get destroyed in NU.
What were its main roles when it was/is used?
Sawk is mainly used as a great wallbreaker and sweeper with its Choice Band set coming off a powerful ass 125 base Attack stat along with a really good 85 base Speed stat allowing it to outspeed the majority of an opponent's team and destroying common walls such as Quagsire. It can also perform other roles such as preventing SR sometimes with Mold Breaker breaking through the Sturdies of Golem, Probopass, and Bastiodon and OHKOing them. Scarf is seldom used but makes for a decently powerful cleaner and Life Orb sets were really common during the rounds Braviary was suspected in (spl era) because of the ability to switch up moves and use Knock Off / Toxic / Bulk Up to cripple common counters such as Musharna or Alomomola.
What caused it to have such a significant impact? (Think Movesets, abilities, base stats, etc)
Sawk caused such an impact because it can literally destroy any team without say a Ghost-type or a bulky Psychic-type such as Misdreavus or Musharna. Its pure power is why common Pokemon such as Musharna, Golurk, and Misdreavus are so common today - to deal with Sawk. Stall teams are known to carry multiple checks and counters such as Alomomola, Misdreavus, and Garbodor all on the same team and can still struggle with Sawk. It can also destroy many balanced or offensive teams because a lot of offensive mons are OHKOd and outsped such as Samurott or Ludicolo and its the reason why outspeeding the base 85 Speed tier is so important for a long time.
How would you deal with this Pokemon when you are team building/battling it? (checks, counters, etc)
Musharna and Misdreavus are the best counters to Sawk, but if you don't carry them on a team, it is important to have really good synergy and good resists/immunes such as Golurk or Exeggutor to abuse the usual Choice-locked Sawk and to make sure you don't get swept by it. Most of the time Sawk is just there to beat down a lot of walls such as Lickilicky or Regice that harm its teammates which makes Sawk even more devastating especially when paired with Skuntank to trap Musharna or Misdreavus. It is important to never let Sawk get a free switch in or a mon for it to easily come in on as well as making sure you don't play too risky against Sawk such as overpredicting an Ice Punch or something because that can really bite you in the ass.