Why is Daniel Striped Tiger a father while Prince Tuesday's a teenager? Because tigers naturally age faster then humans.
It could also be because Daniel was of school age before Prince Tuesday was even born.
When Prince Tuesday was born, Daniel Tiger was already of school age. By the time Mister Rogers' Neighborhood ended, Prince Tuesday was himself of school age. Compulsory school age is usually six years old, so Prince Tuesday was six when Mister Rogers ended, and Daniel Tiger's age was presumably adjusted to reflect the initial age gap between them. The franchise's time skip lines up to reflect the eleven years between the final episode of Mr. Rogers and Daniel Tiger's debut. That makes Dad Tiger about twenty-three years old and Prince Tuesday about seventeen years old as of Daniel Tiger's Neighborhood.
The Land of Make Believe and all of the characters that inhabit it were once only in the imagination of Mr. Rogers. Many of the characters are children or extensions of characters that inhabited the original Land of Make Believe. As Fred Rogers died in 2003, and for there to be a timeline that extends past the run of Mr. Rogers' Neighborhood, we have to assume that Daniel Tiger's Neighborhood takes place within the imagination of a dead man.
For a more Heartwarming explanation- perhaps The Land of Make-Believe was brought back to life by the love of others who wanted to continue showing people that sweetness and magic.
Chrissie's horse is named Peaches. Daniel is allergic to peaches. Therefore, Chrissie's horse Peaches is to be feared, and Daniel was especially justified in asking his father to hold his hand while he rode her.
He's allergic to the fruit peaches, not creatures named Peaches. Just like if someone is allergic to peanuts, they wouldn't have an allergic reaction to an animal whose name is Peanut, if they have hay fever, an animal named Pollen wouldn't make them sneeze, etc.
Daniel has to take the school's pet rabbit snowball home and take care of him. He's got to take Snowball home in a little cage. Daniel says he's never been to a rabbit's house before. Then there's an Imagine Spot of Snowball and a whole family of rabbits partying together, but then things get back to reality, and Snowball has only his cage. He's apart from his family.