When I add a new Email account, Mail application automatically downloads All messages, but I do not need all the messages to be available on my Mac.
Is there a way to tell Mail app to only down messages of the last 3/6 months?
Are you using POP or IMAP? If you use IMAP, you can say in the Mail-settings to omit the attachments. This will only download the email headers and bodies, without heavily attachments. Maybe this can fit your needs?
No - Apple desktop Mail.app doesn't have a setting to only download a window similar to how iOS does on Exchange.