Concept is a bow that's fired a really, really fast and precise arrow. Probably will need to change the activation cost later. I took reference from split-second for the last part of the ability.

You probably want Split Second's wording in there:

"As long as this ability is on the stack, players can't cast spells or activate abilities that aren't mana abilities."

This might cause some awkward timing problems with players who don't understand who has priority, and when. But if you math out all the scenarios where this would be in contest with, let's say, a Giant Growth, it becomes clear that bow trumps growth no matter who has priority.

As a matter of fact, I'm not even sure it's worth adding the 'damage cannot be prevented' clause, since 90% of the time, the Split Second nature of the card will prevent damage prevention anyway.

There's the weird timing of planeswalkers, and how I can get that loyalty counter during my main phase no matter what you do with the Bow of the Eagle. But the same could be said about Prodigal Pyromancer. Loyalty activation is just weird like that.

What's weird about it? It's a cost. All costs are paid immediately.

That's the weird part. It seems obvious to us, because we understand that costs can't be responded to. I've found it's not something that new players intuit. I've had more than one discussion where I had to explain how a Lightning Bolt can't kill a Jace Beleren on the round he enters the battlefield, presuming I'm cautious about it.

But maybe my opinion of how new players respond to loyalty abilities is woefully tied up in interactions from '05-'06. ;)