Designating databases to certain CPUs
Is there a way to make queries from a database to use only certain core on the host?


One way is to use query hint OPTION (MAXDOP 1).

The problem with above apporach is that you have to hint everything that you don’t want to be limited by the server-wide setting, or use 0 (i.e. unlimited) for the server-wide setting and hint everything that you do want to be limited. 


  Can we make queries from 6 "less" important databases A1-A6 to use only one core (always the same core 0) , and leave queries from other 4 "more" important databases A7-A10 to use rest of the cores (in this case 1-3).


A better way is to use Resoruce Governor. It is available only in Enterprise Edition (Dev edition as well).

Note: Make sure you do proper testing before implementing directly in PROD as I have seen really bad cases where without understanding, Resource Governor is implemented and things go horribly wrong.

If you are running on Enterprise Edition then you can create 


CREATE RESOURCE POOL -- here you can specify min and max CPU allowance. Also by using AFFINITY SCHEDULER, you can bind specific schedulers to your Pool.
CREATE WORKLOAD GROUP USING RESOURCE POOL. -- Here you can specify max_dop=1. 
This MAXDOP setting overrides the server-wide setting and any query hints.


Refer to : Resource Governor in SQL Server 2012 whitepaper for more details.

Aaron Bertrand has explained very nicely at : SQL Server 2012 Resource Governor Enhancement