With the native ICS e-mail client, if you have an exchange account with rules to receive certain e-mails to vertain sub-folder, you will not get notification when new mail airrives in those sub-folders. it only notifies you for the "Inbox". Furthermore, the search on server is not optimal and mails are limited to only 30 days.
Regarding your first point, you can get notification for subfolders as well. Tap on the drop-down menu at the top, then 'Show all folders', expand the folders, then long-press on the one you want to sync. At the bottom there's 'Sync options'. Change from Manual to Push and that should be that.
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