There is no point in checking AWE on express edition when SQL Server can see 'just' more than 1G memory. Although it says SQl server can take maximum up till 1G on express edition it takes bit more than that.
AWE is a windows feature in 32 bit which allow SQL Server to use memory greater than its VAS limit. VAS limit for SQL server 32 bit running on 32 bit windows machine is 2G and on WOW system is 4 G. Using AWE SQL Server data and index pages can use memory beyond its VAS limit of 2 G. But since in your case max server memory is 1G there is no point in enabling it. And this is the reason I guess why documentation says so.
As to why it is there because MS does not wants to make change. AWE on 64 bit server has also no meaning but this option is there as well.
I would go with standard edition of SQL if I know for sure that AWE is not supported in SQL Express?
Go with standard edition 64 bit and yes you would also need 64 bit OS for that. 32 bit server these days have lots of performance restriction especially in terms of memory and VAS. 64 bit will have 8 TB VAS and enough memory to support. 32 bit server is kind of outdated now ( unless you have grave requirement)