Is it permissible to go with one's husband and daughter to a hotel to spend Eid day with family, where the family will sit in open areas for eating and children's play, knowing that I usually apologize for not going?

There is no harm in entering hotels if they are free of forbidden things such as alcohol consumption, dancing, and gambling. Otherwise, it is not permissible to enter them, due to the Almighty's saying: "And those who do not witness falsehood, and when they pass by idle talk, they pass by with dignity." So, whoever abandons something for the sake of Allah, Allah will compensate him with something better.