We are building a Cloud-based billing software. This software is web-based and should function like desktop software (Atleast). We will have 5000+ users billing at the same time. For now, we only have just 250 users. We are in a need of scaling now. We are using Angular as a Fronten, Python is used for Backend and React Native for Mobile App. PostgreSQL DB is used for Database. I have few doubts, to clarify before we scale.

Using PostgreSQL for DB will show any issues in the future?

Instead of Integer's primary key, we are using UUID (For easy data migrations, but it uses more space). Is that will introduce any problems in the future?

Do we have to consider any DB methods for this kind of scaling ? (Now, uses a single DB for all users)

We are planning to use one server with a huge spec (for all users). Is that will be good or do we have to plan for anything else ?

Using a separate application server and DB server is needed for our this scenario?
I'll try to answer the questions. Feel free to judge it.
So, you are building a cloud-based billing software. Now you have 250+ users and is expected to have at least 5000 users in the future.
Now answering the questions you asked:

Using PostgreSQL for DB will show any issues in the future?

ans: PostgreSQL is great for production. It is the safe way to go. It shouldn't show any issues in the future, but depends highly on the db design.

Instead of Integer's primary key, we are using UUID (For easy data migrations, but it uses more space). Is that will introduce any problems in the future?

ans: Using UUID has its own advantages and disadvantages. If you think scaling is a problem, then you should consider updating your revenue model.

Do we have to consider any DB methods for this kind of scaling ? (Now, uses a single DB for all users)

ans: A single DB for a production app is good at the initial stage. When scaling especially in the case of 5000 concurrent users, it is good to think about moving to Microservices.

We are planning to use one server with a huge spec (for all users). Is that will be good or do we have to plan for anything else ?

ans: Like I said, 5k concurrent users will require a mighty server(depends highly on the operations though, I'm assuming moderate-heavy calculations and stuff) therefore, it's recommended to plan for Microservices architecture. Thant way you can scale up heavily used services and scale down the other. But keep in mind that, Microservices may sound great, but in practice, it's a pain to setup. If you have a strong backend team, you can proceed with this idea otherwise just don't.

Using a separate application server and DB server is needed for our this scenario?

ans: Short answer is Yes. Long answer: why do you want to stress your server machine when you have that many users.
DB structure for Cloud based billing software, planning issues