Yes. She is still on long nights.
Do they often lay two clutches in a row?

Yes, it's very common for them to keep on breeding or trying to breed for as long as you will let them. If one clutch doesn't hatch they will often start a new clutch as soon as they have given up on the previous one. If the clutch DOES hatch, they will want to start a new one as the babies approach fledging age.

Emma finished with her last batch of eggs the end of June.

After that, I took the paper out that I had over the grate, so she can't chew on it. She is also on 14 hour nights and I restrict their food a bit. So far, there have been no more eggs.

I don't like having to clean the grate, but I am dealing with it. It's better than having eggs.

Emma and Silver hang out together most of the day. But, there hasn't been any mating.

Maybe she's waiting to reunite with Piper.

Emma has gone 8 months without any eggs. I am happy with the success. Of course, spring is coming, so I am trying to keep the nights long in the house.

I think she still pines after Piper. Yesterday, she ended up on the floor under the budgie cage. When I picked her up, she managed to get onto the budgie cage and didn't want to leave.
She ended up on the floor again when I finally shooed her off. Then she went under the cockatiel cage stand and wouldn't come out. She sulked under there for about a half an hour.

She is now 2-1/2 years old. I try to talk to her often, which she seems to like (as long as my nasty hands are nowhere near her). She will take sunflower seeds from between my fingers, but not if my other hand is anywhere nearby.

Sometimes when she gets on the floor and I get her she will climb up my arm to my shoulder. She will stay there a little if there is nothing nearby that she can jump onto. Sometimes she tries to steal my earring. That's something. She is a silly girl. I love her anyway.

Usual for a hen: Bratty when they don't get what they want.

You can try getting her to eat sunflowers from the palm of your hand. That will force her closer to your actual hand, and she will learn to get used to it that way.

It's funny how birds are sometimes. Their motivations can be very silly. Lucy is very task driven, and is always trying to find something to do. Her favorite thing is destroying whatever she can get her beak on, but sometimes she does things for who knows why. For example: On the playstand, she likes to weave the string that has two acrylic beads in between the rungs of the latter it's attached to. I have absolutely no idea why.
Meanwhile Alex is very attention and praise driven. He does whatever he can to get someone to acknowledge him. Lately he's picked up on the habit of causing the most ruckus possible by faking a fright. The way I know he's faking it is because he'll stop after a few seconds and then look to see if I'm watching him. He's learned to "kiss" me (touching his beak against my lips when I say "gimme kiss"), just from me giving him scritches every time he does it.
I'm sure I can use both Lucy's business and Alex's craving for attention to train them both, but I haven't had time to start actually training. Apparently Lucy still recognizes the sound of the clicker, so getting back into the routine shouldn't be hard for her. Alex will most likely catch on quickly.

Sounds like Emma might be somewhat between food and attention driven. She's willing to take sunflower seeds from you, and she likes it when you talk to her. I would make her reach for her food. If you can get her to eat from your palm, try putting the sunflower seeds on your palm, but on a part of the hand that's further away. Eventually, you'll be able to get to the point where she will have to step onto your hand in order to reach it. Every time she does feel brave enough to stretch, give her positive words of encouragement like "good girl" and such. That way, she'll be getting both food and praise for a reward, and that should probably speed the process of her trusting her.

Emma laid an egg yesterday. Could it have something to do with being able to fly again? Nothing else has changed, except for increased masturbation.

I think the masturbating was a huge sign that an egg was coming soon. The flights shouldn't be making her hormonal, unless she was using it to get to places she could use to nest.

I agree with B. Being able to fly shouldn't raise her hormone levels, but masturbating would sure do it. It's a substitute for sex, which leads to egg laying. I can see the progression with my birds - when they first start mating it's infrequent and sometimes not even complete. But mating a little bit stirs up their hormones and they start doing it more, until soon they're doing it several times a day. And that's when the eggs start coming.

Is there a way to discourage masturbation?