Its carries risks
To quickly answer your questions before going into the details:
- would that group be able to breed naturally amongst each other and have fertile children?
Only for if each form of genetic engineering is on a different chromosome.
- would they be able to breed with "regular" humans?
Only if the genetic engineering is isolate to a single chromosome
- If the answer to (2.) is "yes", how likely is it that the offsprings would be fertile?
In a perfect world, 50% chance of genetically altered and 50% chance of normal per trait
For the purposes of this I am going to keep this focused on one example: a genetic alteration giving wings. I will refer to it as the wings gene and have the wings gene traits all be dominant by default. As for whether or not they will be capable of generating enough lift is a totally separate question which I will not cover.
Humans genes are organized into 23 chromosomes that come in pairs. When breeding one of each pair is randomly selected to be passed on to the child from each parent. As such for this wings gene it needs to not only add wings to the human, it needs to modify the existing bone structure on the back of the human for the wings to lock in, greatly increase the human pectoral muscles (and modify them so they can be linked to the wings), and modify the human to be as light as possible. That is a lot of modifications, and if they are spread out and directly integrated into the respective chromosomes that govern those aspects of the body then it will not be as complex and thus reduce the chance of the alteration indirectly harming the individual.
However if that person was to breed with a normal human the resulting hybrid child might end up with oversized pectoral muscles and no wings, be normal weight and not able to generate enough lift, or could be missing part of the bone modification to the back resulting in the shoulder sockets becoming mutilated. None of these are desirable.
As such all these modifications would need to be on a single chromosome so that it will be all or nothing. Since you cannot directly tweak the existing dna on the other chromosomes, you will be having to add additional DNA that suppresses the normal behavior and development of the body that is present on the other chromosomes, much like how the Y chromosome suppresses the development of mammary glands in males (most of the time, and it does not stop the formation of the nipples). That is a lot of extra DNA and the problem is fitting it all on one gene. There are two options for this:
Have one copy have a normal chromosome paired with a wing gene chromosome (similar to how the X and Y chromosome work). This has the benefit of keeping the total amount of dna in the cell roughly the same. The problem is that it would behave like the Y and X chromosomes do without the safe guard that they have (two males cannot breed together and thus cannot result in a YY pair). This is bad since a genetically altered person could only safely breed with a normal person to ensure that they get the normal gene that can be paired with the winged gene. If two people with the wings gene were to try and have a kid there would be a 25% chance of the fetus having two wings gene which would be fatal or carry severe disorders.
Keep the normal DNA and add all the wings gene DNA to it. Thus if they have two copies of the normal + wing DNA they are still good. So anyone with the alteration can safely breed with anyone who has it or anyone who does not have it. If a hybrid was to breed they would have a 50% chance of passing the wings gene along (and two hybrids would have a 75%). Sadly though, this carries its own problems.
The size of the nuclei in the cell is not changing, but now there is more DNA in it. This will increase the chance of something going wrong during cell division. This can possibly lead to health issues, with the worst cases being cancerous cell growths. If this happens early on in a pregnancy or with the sperm or egg then it will likely result in a miscarriage or severe birth defect. There are lists of various genetic disorders that can be found so I will not go into what they are. These lists do help with world building too, if the wings gene happens to be on chromosome 5 then does it increase or decrease the chance of Cri du chat syndrome? How much of a chance these problems have of going up I cannot say, but since this is your world you can decide how much of a factor you want it to be.
As such I recommend for it to be believable and get what you want, each alteration is coded onto a different chromosome and preserves all the normal DNA on said chromosome. Any genetically modified human might have these cool amazing traits, but they also are more prone to cancerous growths, miscarriages, birth defects, and genetic disorders.