I tried to dig out eggs but couldn't find them easily. Then I had to remove those 2 eggs out of the box, so I could dig thru the whole content of the box - found them finally. I thought I wouldn't like to do it again, so I just put all the eggs together on top of bedding. Later she added one more, so total now is 6.

I don't like the situation. The chicks are sitting on parents' cage. I come and try to do something with the chicks (play, communicate), then parents freak out, angrily look out the nestbox, and leave it.

Sorry, no experience to share but just wanted to offer my sympathies. This all sounds hard and with the holidays and a family and anything else, it just sounds... hard!

Thank you Couldy skies, I think its improving a little bit (parents are getting used that nothing is threatening the nestbox I think). Chicks like to climb on parents' cage (because this is where the parents are) and even get inside the nestbox sometimes that causes parents to get out.
I wonder, the more they are flock birds, the more difficult it is to make them companion birds, right? Today it was a funny moment, the hen landed on my head and chicks tried to do the same but they didn't have enough room and kept falling off (at some time I had 4 birds trying to balance on my head ). Interesting, they are cautious to land on me by their own but they would do what mom does. Still babies, I guess. 10 weeks old today.

The eggs still need to be incubated pretty much constantly. It's up to the parents to work out the division of duties. The father tends to feed the older chicks more than the mother does, so until they are weaned the hen might sit on the eggs more in the daytime than she did with the first clutch.

Everything is rather calm. The parents are incubating pretty much all the time, and they are not jumping out of the box with every little noise anymore. The last time I saw eggs there were 7 of them, so I guess all my actions increased their number, not decreased. Last time, when everything was unchanged, she laid only 6 eggs.
I don't know why, but Cuddles doesn't take bath anymore. I always prepare warm bath for her (she likes to bath) but instead she gets inside much smaller shallow water dish that has cold drinking water. I wonder if she prefers cold bath to the warm bath. Trillie (and the chicks) like to be misted from the spray bottle.
I will candle eggs in a few days if parents cooperate and let me.
The chicks are doing fine, I think. Their weight is between 85 and 92g. I don't recently have a chance to feed them formula. Every day I am planning to but something gets on the way.

The chicks' weight looks fine. If you haven't handfed them for several days and their weight is staying steady, it means they don't need to be handfed any more.

I candled the eggs. Almost all of them are fertile and there are 8 of them! (If I understand right what fertile look like), but some also looked like there were just laid so its hard to say yet. I mean, A LOT of fertile eggs! I hope the hen would stop laying them now (well she can't count to 8 and she didn't read in the internet that 8 is maximum for tiels, so I am a little worried), I lost a track of her dates. This is the hugest clutch for us so far.

She also looks tired. Often standing still looking at one point being deeply in her thoughts. I don't know if it's normal.

They incubate eggs well.

The chicks got a habit to land on my head and back (especially when its feeding time in the morning), they try to do it all together but some have to leave due to high competition.

I've heard of hens having bigger clutches than that, but there's a good chance that she's through. It seems like some hens want to lay a specific number of eggs inside the nestbox, and any eggs that were laid outside the box don't count, even if you move those eggs into the nest. Your girl might be one of these.

Laying eggs is hard work, and a hen will look especially tired if she laid an egg within the last couple of hours. Make sure she has a nice healthy diet and let her rest when she wants to.

thanks.
Tielfan, what do you think about using formula past 30 days since opening? Do you follow this recommendation or not really?

I don't think my handfeeding formula had that kind of instructions on it. If you can tell me the brand name I can look for information about it on the internet. I usually keep most of the formula in the freezer, and a small container in the refrigerator that I use to actually mix up the formula. Cold storage will help preserve the nutrients longer.