If I delete something from my Boardmaker Account will it be deleted from Boardmaker 7 (BM7) as well?

Yes. Those two locations are automatically synchronized and if you delete an activity from either, it will be deleted from both.
For more detailed information about how this works please see the information below.
When you create an activity in BM7 and save it, it will by default save to a Boardmaker folder on your local computer. That folder mirrors what is stored in your My Boardmaker account. While BM7 is running and you are signed into your My Boardmaker account these folders are kept synced automatically.

1. ­If you create/save an activity or folder while using BM7, the file/folder shows up in the My Boardmaker area of the dashboard, and automatically syncs to the activities list in your My Boardmaker account.

2. Likewise, if you create a folder or add a file from the community to your My Boardmaker account while on the My Boardmaker website, those changes will be reflected in the BM7 dashboard.  The only difference being that the files will not be downloaded until you click on them.