Applying a scrum to a small startup

Dog trainer is a new one on me but I love it and there's no reason why not.

1. Is it right for the members of the scrum team to continue the project together without changing the members unless there is something special once it is decided?
Or, can the scrum members be different for each project?

In Scrum the most important project is the Sprint, because that's the one which allows empirical process control to be established and maintained. If team membership must change each Sprint to ensure the Sprint Goal is met with an empirical outcome, then team membership may change accordingly. The important thing is for teams to self-organize this for themselves, and to recognize that there may be a drop in productivity until a new member beds in. Stability is valued for that reason.

2. If team members rarely change, do you keep team members the same only at the request of members? Or do I need to change my team members periodically (e.g., quarterly)?

You don't change them at all. Team members are not there to have things done to them by others. Self-organization is the key. It would be better to sponsor that approach and its outcomes.

3. If we're building a team, we're considering creating two teams with one backend developer per team, and UXUI designer and full stack developer participating in each team at the same time. 
In this case, the UXUI designer and the full stack developer will be on both scrum teams at the same time, so can I make a team like this?

You can't. They can. Help them to understand and explore the implications, so they can take ownership of the challenge and self-manage the situation most effectively for themselves.

If the above idea is wrong, how is it best to make a team?
(For your information, two backend developers can also do some front-end development.)

Why not create a bounded environment for them to self-organize within. Timebox it, with a clear goal (e.g. each team must be able to complete a Done and valuable increment of usable work) and rules (e.g. no more than X people in each team). Facilitate the workshop, all the time seeking to reveal rather than resolve.

4. I think the CEO will play the role of PO. Does it matter if one person is in charge of multiple teams' POs? 

The Product Owner is accountable for value. Why would a PO need someone "in charge" of them at all? 

5. As far as I know, the sprint planning meeting is not conducted across the company, but by teams. However, if we proceed as in the above case, how can UXUI designers and full stack developers participate in the sprint planning meeting?
Should the Sprint Planning Meeting be held at different times, so that designers and full stack developers participate in each team meeting?
Or should we get together across the company and have a Sprint Planning meeting?

Anyone who will be doing work to meet Sprint Planning commitments is a Developer and will need to be there.

How would they choose to self-organize their working patterns so these joint commitments can be framed and met?

6. I think this is a little silly question. Operations team members such as marketers, dog trainers, and content editors rarely participate in app development.
In this case, does the development team work in an agile way and the people in the operation team work in a different way?
Then, you don't need people from the operation team to participate in the Sprint Planning meeting, do you?

Perhaps not, but they could be invited if they have useful input to provide. They might need to be more actively involved in helping to refine Product Backlog items so they are then ready for Sprint Planning, such that the Developers are confident they can take on and complete that work.