What are valid connection strings for SSMS login box?
There are various possibilities and I will try to answer them according to the numbered questions.

Valid connection strings for the SSMS login box are:

1st instance


Using the host name


SERVERNAME 
SERVERNAME,1433

Using the CNAME/Alias


SERVERNAME-I01 
SERVERNAME-I01,1433

Using the IP address


123.1.2.3 
123.1.2.3,1433

For the Dedicated Admin Connection (DAC); depends on Browser Service status

a) Browser Service ON 


ADMIN:SERVERNAME
ADMIN:SERVERNAME-I01
ADMIN:123.1.2.3  


b) Browser Service OFF   


SERVERNAME,1434
SERVERNAME-I01,1434
123.1.2.3,1434  



2nd instance

Because the 2nd instance has a dynamic port for the SQL Server instance and a dynamic port for DAC, the ports have to be retrieved from the ERROR log file in advance before connecting to the SQL Server. See 7 for more information. In this example I will assume port 63390 is used for the SQL Server instance and port 63389 is used for the Dedicated Admin Connection. Microsoft recommends to fix the IP port, thus allowing you to properly configure a firewall 6.


Using the host name


SERVERNAME\Instance2 (Browser Service ON)
SERVERNAME,63390 (Browser Service OFF)

Using the CNAME\Alias


SERVERNAME-I02 (Browser Service ON)
SERVERNAME-I02,63390 (Browser Service OFF)

Using the IP address  


123.1.2.3,63390 (Browser Service is ON or OFF)
Using the IP address without port, would connect you to the default instance  

For the Dedicated Admin Connection (DAC); really depends on Browser Service status  

a) Browser Service ON


ADMIN:SERVERNAME\Instance2
ADMIN:SERVERNAME-I02\Instance2 
ADMIN:123.1.2.3\Instance2
Always requires instance name, because the CNAME or IP address references the same IP address as the default standard instance  


b) Browser Service OFF  


SERVERNAME,63389
SERVERNAME-I01,63389 
123.1.2.3,63389  



3rd instance


Using the host name  


SERVERNAME\Instance3
Because this instance is using the same SQL Server port as the default instance, but a different IP address, the SERVERNAME,1433 notation would not work. You would be connected to the default instance.

Using the CNAME/Alias


SERVERNAME-I03 (Browser Service ON)
SERVERNAME-I03,1433 (Browser Service OFF)

Using the IP address 


123.1.2.4 
123.1.2.4,1435

For the Dedicated Admin Connection (DAC); depends on Browser Service status

a) Browser Service ON  


ADMIN:SERVERNAME\Instance3
ADMIN:SERVERNAME-I03 (works because the CNAME references a dedicated IP address)
ADMIN:123.1.2.4   


b) Browser Service OFF  


SERVERNAME,63389 (works because the port references a dedicated port for DAC)
SERVERNAME-I01,63389 (works because the CNAME and port reference a dedicated instance/DAC)
123.1.2.4,63389  





Browser Service

If the browser service is DISABLED or has not been started, then connections cannot be routed to the relevant port. In this case the DAC port has to be retrieved from the ERRORLOG file of the instance. You should find a line with something like Dedicated admin connection support was established for listening remotely on port 63389. for the DAC port. This port number is the port the DAC will listen on. You are required to supply the relevant port for the Dedicated Admin connection and the corresponding port for the SQL Server connection if your Browser Service is turned off. You do not have to add the instance name to the connection string when using the port, because the port is reserved for the instance.

In short the Browser Service is responsible for converting instance names to ports. If the Browser Service is not running, then you have to supply the port instead of the instance name.



Reference Material


Logging In to SQL Server (Microsoft | Docs | SQL Server)