New Kitty vs Old Kitty: What is playing, what is fighting, and what to do??

As you probably know, I brought a new addition to the family, Bugsy, last Saturday, so it has been 5 days.
There was no official introduction; I left him for a few hours in the bathroom, but he was crying so much, and when I opened the door he sneaked out... I decided to not lock him up anymore, since both of them are mellow ragdolls...
He has, for my surprise, brought the best out of Lucky. Since he came in Lucky has been eating more, playing more, and overall more active and cuddly with me. For the most part their interaction is pretty peaceful, unless they get startled, then they react...
For example: Lucky and Bugsy were sleeping on the couch last night, very close to each other, pretty much snuggling... When I got out of the couch, Bugsy got startled, and bit Lucky on her back twice. She ran away, and I checked on her, all is fine; no skin has been broken or anything...
Today he jumped on the bed and startled her, she put her paw up and hissed at him - 1st hiss in five days!
They hang out under the bed together, and both are sleeping on my bed with me.
Lucky watches Bugsy VERY closely, and she keeps him "in line" He will jump on the counters, she will fight with his tail to bring him down... Fun to watch!
Here is my question: Of course Lucky is establishing her hierarchy, and I want to allow her to do that, I just want to know what is normal and healthy, and the best way to intervene when necessary.
There is a lot of chasing, and I don't know when they are playing or being mean to one another. Lucky chases Bugsy, then they turn around and Bugsy chases Lucky. Is this chasing normal?
How do I recognise a fight, and how do I respond to it?
Sorry for the lengthy post - I never had more than one cat at a time before, and I am lost on how to protect them...
Thanks!

Sounds like things are pretty okay. If your cats are not getting along, they would not be sleeping or snuggling together. The chasing is normal. Running, pouncing on each other is part of the fun.

It all sounds perfectly normal. Kitties chase each other. They have little chase and ambush routines. They "bite" each other. They hiss to say "enough is enough".

One way to easily tell whether they are playing are fighting has to do with how easily you can interrupt them. When my boys are playing, I can easily get their attention. If they are fighting, one or both remains focused on the other when I try to draw their attention away by calling their names or clapping.

Chasing is normal, especially when they take turns. If one of them was always doing the chasing and the other was scared, then it could be a problem.

Sounds like things are going very well! My two chase each other around all the time --as others have said as long as it is not always one stalking/chasing the other, you're OK. Sleeping in proximity is a very good sign.

I would expect some wrestling matches as they determine who's who. I would also be prepared for their relationship to change over time--we have a basically peaceful household but from time to time I'll notice that they're having a spat...it will pass

My Ragdolls love to chase each other through the house. Only 3 out of the five actually growl and hiss. None of them ever has drawn blood. Your two are playing the pecking order game between them. I honestly never worry about my Ragdolls as they will be sleeping next to each other 30 minutes later