This is a really cool item.

The ability to send it to any familiar place within a mile is super handy. Let the owner fly up to the top of a mountain or tower, send the broom back to pick up the rest of the party and bring them up two by two (depending on their size) No reason people couldn't hop on. They wouldn't be able to control it but it's a super handy feature. Also could do deliveries and send messages or packages, or other useful effects.

Handy notes: it takes 105.6 turns to travel a mile at 50ft per turn. That's 10.56 minutes for each mile.  Every 10 feet of movement is 1.136363636363636 miles per hour. 

Two questions:

1. Can you take a dash action on a broom of flying?

2. How do you "ride" it? Do you need the vehicle proficiency? Is it an acrobatics or Dexterity check? Does it take any skill or is it simply magical?

Dm discretion probably

for 1) I'd almost certainly allow this, as the Dash action means that whatever portion of your turn you'd have spent not moving, you're spending on movement instead.

For 2) I think this is entirely up to the DM's desire for flavor. In my games they can use it for basic travel without difficulty, but complications (stiff wind, etc.) often call for skill checks.

If you attach a dagger to the brim of flying could you command it to impale someone

DM discretion... i'd allow it a few times

No you cannot take the Dash Action with a Broom of Flying. It has a set speed.

I would say that the target would get a fairly low DC Dexterity saving throw, as they would be dodging out of the way, rather than you wielding the broom like a spear (which would be an attack roll). Maybe DC 12 or 13?

I completely forgot about this and it has just made my day that you replied 

 What would be the cost be?

My broom of flying fell in some fire for two rounds of combat. Will it survive the fire?

Magic items are supposed to be very difficult to destroy. What is the hp of the broom?

Anyone up for a game of quidditch?

If I leave my broom in my house, but I am over the 1 mile distance, could I open a portal to my house and call the broom through it?

50 feet per 6 seconds is 8.33 feet per second, or 5.68 miles per hour.

Quote from CaptainKyrsi >>

50 feet per 6 seconds is 8.33 feet per second, or 5.68 miles per hour.

 Exactly! 1.136363636363636 x 5 = 5.68

If I leave my broom in my house, but I am over the 1 mile distance, could I open a portal to my house and call the broom through it?

I'd allow it. Since you would be using resources i.e a spell slot to access the broom. 

A would think that the distance would be determined by the most direct path ignoring obstacles. With how portals shape space, a path that passes through the portal (assuming the distance between the broom and the portal and the portal and you were less than 1 mile total) would be less than a mile even if without the portal the distance would be greater.

A dungeon master who argues that a broom you can reach out and grab through a portal arguing that the broom is more than a mile away must believe you have really long arms.

this can derail a campaign

I would be careful with dividing the party like this. You can have them all be attacked at the same time. And they are far from each other.