Yes, Dash will double the flying speed of the pegasus, allowing it to fly 180 feet in a round.
Horseshoes of Speed only increases your walking speed, so it won't help your flight. Haste will work though, so a Hasted Pegasus can fly 360 feet on its turn if it dashes.
Yes. A question I don't know the answer to but most DMs say yes to is the same question for objects with a Fly speed - action economy isn't well-defined for objects.
Yes, if you want to go straight down. It's called "falling". You can go 500 feet down instantly. That's really fast. Be careful though. That 20d6 damage at the end isn't pleasant.
Is there any way that I can increase the speed faster than 360?
Geann is right, Xanathar's has optional rules letting any flying creature mitigate falling damage or arrest a fall. You'll need to be very high up - the pegasus has to go prone to stoop like this, which means it falls on its turn, and then on its second turn, it can stand up, fixing the prone condition and arresting the fall.
EDIT: I just went over the Xanathar's rules, and they don't make any sense - there's no explanation why you'd need to wait until turn 2. If your pegasus goes prone and falls while it still has at least 45 feet of movement left, it can spend 45 feet to stand up from prone, which should end the fall.
Also, your 360 number is wrong. Horsehoes of Speed are +30, not x2, so if they worked on flying speeds, the pegasus would be 120 with them, 240 while dashing. But Horseshoes of Speed only apply to a creature's walking speed. What you're thinking of are Boots of Speed, which also only apply to walking speed.
If a Pegasus can dash, can she double her speed from 90 to 120? And if Horseshoes of Speed are used can haste also be cast on her.
Yes, Dash will double the flying speed of the pegasus, allowing it to fly 180 feet in a round.
Horseshoes of Speed only increases your walking speed, so it won't help your flight. Haste will work though, so a Hasted Pegasus can fly 360 feet on its turn if it dashes.
Yes. A question I don't know the answer to but most DMs say yes to is the same question for objects with a Fly speed - action economy isn't well-defined for objects.
Is there any way that I can increase the speed faster than 360?
Yes, if you want to go straight down. It's called "falling". You can go 500 feet down instantly. That's really fast. Be careful though. That 20d6 damage at the end isn't pleasant.
<Insert clever signature here>
Geann is right, Xanathar's has optional rules letting any flying creature mitigate falling damage or arrest a fall. You'll need to be very high up - the pegasus has to go prone to stoop like this, which means it falls on its turn, and then on its second turn, it can stand up, fixing the prone condition and arresting the fall.
EDIT: I just went over the Xanathar's rules, and they don't make any sense - there's no explanation why you'd need to wait until turn 2. If your pegasus goes prone and falls while it still has at least 45 feet of movement left, it can spend 45 feet to stand up from prone, which should end the fall.
Also, your 360 number is wrong. Horsehoes of Speed are +30, not x2, so if they worked on flying speeds, the pegasus would be 120 with them, 240 while dashing. But Horseshoes of Speed only apply to a creature's walking speed. What you're thinking of are Boots of Speed, which also only apply to walking speed.
Haste lets you take an additional dash, so you could have a speed of 540.