What would be good alternatives that could replace "beautiful" in this context?

Can it be "when taking the shot" instead here?

Good question! You could say it, but it would imply something slightly different. Here, "to take" means the possible moment of taking a picture, whereas "taking" would mean within the process and action of taking a picture. So, if they said "when taking the shot," it would mean that while they were taking the shot, they weren't sure about something. "When to take a shot" means the right moment or time to execute the action. Ex: When editing photos, I like to make sure it's the right size first. Ex: I never know when to edit photos during the week. Ex: I hit my head when I was walking downstairs. Ex: I'm not sure where to walk so I can avoid the puddles.

I thought the place that children go instead of school is called "preschool". But is it more common to say "play group" in British English? What's the difference?

Good question. "Playgroup" is a group that gives children under the age of five in the UK, the opportunity to play and learn together. We do not use the word "preschool" in the UK, it is an American term. "Preschool" in the UK would be referred to as "Nursery". Playgroups are less formal than a Nursery school. They do not provide full-time care and usually only open for a few hours a day during school time usually in the morning. As such they are more comparable to American "daycare". Ex: Her playgroup is only until 1 pm. Ex: We can go swimming after playgroup.

What would be good alternatives that could replace "beautiful" in this context?

To express the same feeling of admiration, the words "gorgeous," "amazing," "handsome," "wonderful," and so on can all be used. Any adjective that expresses a strong sense of admiration (especially in regards to appearance) can be used here. Ex: Wow, what a handsome greyhound. Look at its lean body. Ex: My friend has a gorgeous Maine Coon cat.

I thought "come from" is used when people talk about their nationality. What's it mean here and is it used in other contexts too?

Good question! "To come from somewhere/someone/something" means "to be from ______(place, person or thing)". It is used to find out the origin or source of something (for example, someone's nationality is their origin). Max wants to know the source/origin of the tickets. In this context, the source is not a place but a person (Moon). Ex: "These tickets come from Mary!" Ex: "Mary comes from Sydney, Australia."

Is it necessary to put "to" here? Can I just say "I need you come with me"?

Yes, the preposition "to" is necessary in this context. "To" here connects the pronoun "you" and the verb "come" together, and show relation between these two words. "To" is also used here to express purpose. Without "to", this sentence would sound unnatural. Ex: I want you to help me clean the house. Ex: She is coming over to eat dinner.