The last government brought in a law saying that classes in Reception, Y1 and Y2 can have a maximum of 30 children with a single teacher. There are only a few cases where classes are allowed to go over that limit legally. The only one of those that is relevant for appeals is where there has been a mistake and the child should have got a place. You may not like it but I'm afraid that is the law.
Many schools simply don't have the space to open up an additional class. Some people object to the idea that popular schools should be allowed to expand, leaving other schools half empty. Of course, the school may not want to expand and parents of children who are already there may be very unhappy to find the school putting up extra classrooms and taking in additional children.
Regarding your grandson, they must listen to an appeal regardless of the reasons. They cannot simply reject an appeal without a proper hearing. However, if it is an infant class size case you should only win if a mistake has been made (that's the three categories your daughter has been given). Typically that means your grandson has been placed in the wrong admissions category or they have measured the home to school distance incorrectly. Check to see if this may have happened and check the admission number (PAN) - that will confirm whether or not it is infant class size. If you are unsure come back here and we will advise.
The chances of a successful appeal would be better if it isn't an infant class size case but even then the difficulties you outline for your daughter's work don't make a compelling case. Appeal panels simply can't admit every child where the parent claims that it will cause them problems at work and/or mean they will have to leave their job if the child is not admitted. Schools wouldn't be able to cope with all the children.
If it is not an infant class size case and you can show that your grandson's welfare would be damaged by not being admitted to this school you would stand a reasonable chance of success at appeal. However, being shy is not a good reason for admitting a child unless you can produce expert evidence showing that your grandson is particularly badly affected and that, in the opinion of the expert, he needs to go to your preferred school. Similarly with problems going to the loo.
Unfortunately it sounds like they are saying this would be an infant class size appeal in which case you need to look for a mistake.