1. If they want to lay an egg, are their any signs before that?
Typically, a hen will become more hormonal... that is, she may start shredding things up more and looking for a place to nest. May become "trigger happy" with her beak, too.
2. If they beginning to build a best, should I allowed that or not? (but perhaps it is the male that building nest?)
Typically, it's the females, and that's entirely up to you. I discourage breeding behavior and if I have a hen that starts laying eggs, then I do what I can to stop the behavior. Some will argue that it's a natural behavior, of which I agree, however nothing is natural about living within captivity... and there are some hens that will continue to lay eggs and never stop, or birds who are so ramped up on sexual hormones that they don't know how to calm down. They just become a hot mess. It's pretty bad when a complete stranger can go up to a bird and that bird starts displaying sexy behaviors and trying to regurgitate because that's become the only way they know how to interact with humans.... to get sexy. To me, that is not healthy! It is not healthy for a bird to continually desire to mate/breed with humans. It can be hard to determine whether or not it's safe to allow sexual/hormonal behavior until it happens.
I've never done anything drastic to stop it.... I mean, the most drastic thing I've done to discourage egg laying was moving a flock of birds from one cage to another and taking the cage out of the room! I've never had to do hormone (Lupron) injections, spay, or do the implant.
3. How can you tell if the egg gets stuck? And I want the first signs of that.
This can be difficult to determine... but main signs include straining, difficulty breathing and unable to go to the bathroom.
Hens can display this behavior prior to laying an egg, but they can also live with an egg inside of them. If a hen never lays an egg that they developed inside of them (I have heard of 3 cases - 2 cockatiels and 1 budgie), then the egg becomes calcified. Basically, a well cooked egg with tons of layers of calcium! Most hens who can't lay an egg usually become egg-bound.
4. If the last thing happen, is their anything I can do?
You can attempt to take steps to relieve discomfort, such as feeding olive oil or coconut oil, rubbing it on their cloaca, giving them a warm (but not hot!) 'bath' to attempt to relax the muscles... Definitely something to read up on!
And speak to an avian vet about as well!
5. How urgent is it if the egg gets stuck?
Unfortunately, I don't think there is any specific time-line of when you need to worry, unless you notice obvious distress signs from a hen. That is, if your hen is acting normal, just keep an eye on her! If her behavior doesn't look good, seek the advice of an avian vet!
It is possible for an egg to break inside of a hen and this could result in the death of a hen. Likewise, if a hen can't lay an egg, one procedure a vet can do is to drain the egg (while inside the hen) then break the egg itself and wait for her to pass the remnants naturally.
6. Is to remove the eggs and replace with dummy eggs the best thing to do?
Really depends on your hen. I've had birds for nearly 17 years now and I've never had fake eggs. I have cockatiels, *notorious* for being chronic egg layers. I do not have chronic egg layers. Not even from a hen that was going down that route before I took her in!
7. For how long should I accept that they lay on the egg? if they not are eating or drinking at all, and don't accept food and water from me.
Eggs. How long? At least 3-4 weeks, if not longer... if caged together, then the male should at least be feeding the hen. If not, the hen should at least be leaving the nest to get food and water, although she may not get enough for herself. Again, depends on the hen.
Considering your birds are flighted and they eat well, I highly doubt that egg binding would be an issue. It is possible, but it's more likely to occur in hens on a poor diet, that are over-weight, don't get enough exercise.... granted, some hens are prone to laying eggs that are physically too large to lay, in which case it doesn't matter how well they are cared for, they're screwed regardless... and need medical assistance to remove said egg. I've only heard of this happening in ringnecks, cockatiels and, I think, finches, so far. It may have happened in other species that I'm not aware of.
And also to Monica for that great pics and description. 
Only sad that he never told how they tasted. 
