You need to decide which options below you want to implement. Here are the options below. Based on your questions i think you want to implement Hot standby disaster.
Cold standby. A secondary data center that can provide availability within hours or days.
Warm standby. A secondary data center that can provide availability within minutes or hours.
Hot standby. A secondary data center that can provide availability within seconds or minutes
Cold standby disaster recovery strategy: A business ships backups to support bare metal recovery to local and regional offsite storage regularly, and has contracts in place for emergency server rentals in another region.
Often the cheapest option to maintain, operationally.
Often an expensive option to recover, because it requires that physical servers be configured correctly after a disaster has occurred.
Cons: The slowest option to recover.
Warm standby disaster recovery strategy: A business ships backups or virtual machine images to local and regional disaster recovery farms.
Pros: Often fairly inexpensive to recover, because a virtual server farm can require little configuration upon recovery.
Cons: Can be very expensive and time-consuming to maintain.
Hot standby disaster recovery strategy: A business runs multiple data centers, but serves content and services through only one data center.
Pros: Often fairly fast to recover.
Cons: Can be very expensive to configure and maintain.
If you choose Hot standby recovery then you need to setup the following:
- A separate configuration database and the SharePoint Central Administration website content database must be maintained on the failover farm.
All customizations must be deployed on both farms.
Operating system, SQL Server and SharePoint products software updates must be applied to both farms, to maintain a consistent configuration across both farms.
You can copy SharePoint products content databases to the failover farm by using asynchronous mirroring, asynchronous commit on an availability group replica, or log-shipping.
Service applications vary in whether they can be log-shipped to a farm. For more information, see Service application redundancy across data centers later in this article.
The following question assuming that you choose Hot standby option.
Does CA database and Configuration database need mirroring?
Yes, A separate configuration database and the SharePoint Central Administration website content database must be maintained on the failover farm.
Does Site collections will be automatically available on the DR farm for users to access, or only the data of site collection will be mirrored and I have to do some extra steps to make it available?
You need to create the sitecollection since it is in different farm topology.
Does custom web parts will be automatically available on DR farm, which means that I just need to re-add them to the desired place
Yes, you need to deploy any latest custom solutions.
Does User Profile database and Search Content database need to be mirrored? If yes, what is the additional configuration steps to make them work on DR farm
You need to create and configure user profile services and database and search in this farm as in production environment.
Therefore if the accident happen, the DR Hot standby will have exactly the same like production.