My test of whether a hen is broody enough to deserve eggs is that she has to spend two consecutive nights on the nest instead of where she normally sleeps. I don't care what she does during the day or how she acts during the day. Does she spend two consecutive nights on the nest?
Please make your decision on weather you want to let her sit based on if you want chicks now or not. Her "unusual" behavior sounds like more wild instincts. I have had several pullets brood eggs just fine.
I agree with Yardmom. I've had as much success with first time broodies as I have with hens that have actually hatched and raised a brood before. I've had as many problems with broodies that have hatched and raised before as I have with first time broodies. I don't give guarantees about chicken behaviors, you never know what any one living animal will do in any specific case. I don't know if she will be successful or not but if I wanted her to hatch chicks I'd give her eggs.
My Question : How long can an egg sit after being laid and still be viable without her sitting on it? That question is because I'd like to slip some EE eggs under her but I work during the day so it would be like 5 hrs before I can do that.
A broody hen will try to hatch her own eggs, any other hen's eggs, duck, turkey or pheasant eggs, fake eggs, door knobs, and about anything else. So you can give her other eggs.
Under ideal conditions you can store eggs two weeks or more before you start incubation and expect them to hatch. Most of us don't have ideal conditions but you can still store them a while. The further away from ideal conditions the shorter time you can store them. People have written very long articles about this and it can get complicated but I'll give you a short simple method. Lay your eggs on their side in a place that stays under 80 degrees Fahrenheit. Put an x on one side of the egg and an o on the other side. Three times a day roll the egg over. Do this and your eggs should be good for at least week.
All the eggs need to go under her at the same time. If you don't start them at the same time you get a staggered hatch where the hatch is spread out over several days. Those are seldom good hatches and are very stressful.
To keep her broody, if she truly is broody, mark an egg or two and leave them under her. These are sacrificial. Other eggs may show up under her if the other hens have access to the nest. As long as you remove those new eggs daily you can still eat them.
You have two options. You can let her hatch with the flock or isolate her from the flock. If you let her hatch with the flock, collect all the eggs you want her to hatch and mark them. I use a black Sharpie. Then when you have all the eggs you want her to hatch remove the sacrificial eggs and throw them away. Put all the hatching eggs under her. Then once a day after the other hens have laid check under her and remove any eggs that don't belong.
If you isolate her, build a pen with a nest and room for food and water. Build it so she cannot get out and no other hen can get in. Plan on leaving her in that until the eggs hatch. She may break from being broody when you move her so move the sacrificial eggs with her until you see that she has accepted the move. Then replace the sacrificial eggs with the real hatching eggs.
As for her sleeping in the tree. If she is not broody, if you lock her in the coop for a week or more, she will likely get in the habit of sleeping in the coop. That's the only way I can think of to retrain her, don't let her sleep in that tree for at least a week. Even that may not be long enough.
Good luck. Let us know what you decide and how it goes.