After a forced failover testing with async mode, do I need to rebuild my old primary database?
I'm not sure exactly what you mean by "rebuild" the database, but provided the databases are still in a working condition then you shouldn't need to take any actions like that.
What you're seeing, by performing a forced failover, is by design. If you do a forced failover, you could potentially have failed over to a replica that isn't completely caught up or at the same point-in-time as the primary replica. Because of that, data movement is suspended to the secondary replica(s) from the now primary replica so there is a way to have manual intervention if you are now on a database that is "behind". That behavior that you are seeing is a good thing.
This BOL reference explains it all:
After a forced failover, all secondary databases are suspended. This includes the former primary databases, after the former primary replica comes back online and discovers that it is now a secondary replica. You must manually resume each suspended database individually on each secondary replica.
When a secondary database is resumed, it initiates data synchronization with the corresponding primary database. The secondary database rolls back any log records that were never committed on the new primary database. Therefore, if you are concerned about possible data loss on the post-failover primary databases, you should attempt to create a database snapshot on the suspended databases on one of the synchronous-commit secondary databases.
Please see this BOL reference on how to resume an AG database.
The T-SQL for this would be:
alter database YourDatabaseName
set hadr resume;
NOTE/WARNING/DISCLAIMER: You really need to do some leg work to ensure that you are not causing data loss by resuming data movement. See above, it could be a huge problem. The point of suspending data movement is for this very reason: To manually make sure you can recover as much data as possible. When you resume data movement, that could be irreversible. When you resume data movement after a forced failover, you have to have the words "potential data loss" in the forefront of your mind always.
Does it mean that former primary replica (A) will become primary replica automatically?
No, all that means is when your replica comes back into the picture, and when the Availability Group database gets back in a SYNCHRONIZED state that it would be failover ready. That operation will not happen automatically. You indeed would have to either do this "failback" manually, or engineer a way to automate this (rather simple, let me know if you want to explore those options).
From a high-level view, your listing of steps is complete after step #3.
Best Answer
The short answer is No you cannot do that. First, if you suspend the data movement in one of the availability group database, you no longer able to failover it to your secondary node. Unless you make a forced failover. If you want to suspend the data movement in one of your databases, and also want to make the failover, you have to remove that specific database from the availability group and change the data. However, to add it back to the availability group, first you need to synchronize it back again.