As there are more than one bounded contexts that need to be queried for the validation to pass you need to consider eventual consistency. That being said, there is always a chance that the process as a whole can be in an invalid state for a "small" amount of time. For example, the user could be deactivated after the command is accepted and before the order is shipped. An online shop is a complex system and exceptions could appear in any of its subsystems. However, being implemented as an event-driven system helps; every time the ordering process enters an invalid state you can take compensatory actions/commands. For example, if the user is deactivated in the meantime you can cancel all its standing orders, release the reserved products, announce the potential customers that have those products in the wishlist that they are not available and so on.
There are many kinds of validation in DDD but I follow the general rule that the validation should be done as early as possible but without compromising data consistency. So, in order to be early you could query the readmodel to reject the commands that couldn't possible be valid and in order for the system to be consistent you need to make another check just before the order is shipped.
Now let's talk about your specific questions:
How to know that the customer is really exists in database (query-side customer service) and still active?
You can query the readmodel to verify that the user exists and it is still active. You should do this as a command that comes from an invalid user is a strong indication of some kind of attack and you don't want those kind of commands passing through your system. However, even if a command passes this check, it does not necessarily mean that the order will be shipped as other exceptions could be raised in between.
How to know that the product is exists in database and the status of the product is published?
Again, you can query the readmodel in order to notify the user that the product is not available at the moment. Or, depending on your business, you could allow the command to pass if you know that those products will be available in less than 24 hours based on some previous statistics (for example you know that TV sets arrive daily in your stock). Or you could let the customer choose whether it waits or not. In this case, if the products are not in stock at the final phase of the ordering (the shipping) you notify the customer that the products are not in stock anymore.
How to know whether the customer eligible to get the promo price from the related product?
You will probably have to query another bounded context like Promotions BC
to check this. This depends on how promotions are validated/used.
Is it ok to call API directly (like point-to-point / ajax / request promise) to validate this payload in order command-side service? But I think, the performance will get worse if the API called directly just for validation.
This depends on how resilient you want your system to be and how fast you want to reject invalid commands.
Synchronous call are simpler to implement but they lead to a less resilient system (you should be aware of cascade failures and use technics like circuit breaker to stop them).
Asynchronous (i.e. using events) calls are harder to implement but make you system more resilient. In order to have async calls, the ordering
system can subscribe to other systems for events and maintain a private state that can be queried for validation purposes as the commands arrive. In this way, the ordering system continues to work even of the link to inventory
or customer management
systems are down.
In any case, it really depends on your business and none of us can tell you exaclty what to do.