Can Office 365 also store all emails on local PC

Hi DaveBlue001 

Yes, this can all be done in Office 365; If you already have Office then you can either use Office 365 Business essentials or just Exchange Online if you don't want to have SharePoint/teams etc.

How it works is the mailbox will be hosted by Microsoft on their servers - when you send and email it on your laptop, it'll show the sent email on your desktop (and on the web) so you can see what was sent from anywhere. Same goes for creating/deleting/moving an email - the change is made on the mailbox in the cloud, the Outlook client on laptop/desk will just sync up the changes.

There is an option to run Exchange in 'cached' mode - this will create a special file on the laptop/desktop called a .ost which will have a local copy of emails going back to however long you need - 3 days, 6 months, or everything so if your internet plays up you can still get emails. Once your device has internet again, changes will be synchronized up. 

Will also work will with iPhone/Android - the Outlook app is pretty good imo

Should also be fine for a single address - you can have alias' on the address, but if you want a seperate mailbox it'll be an additional licence. 

https://docs.microsoft.com/en-us/office365/servicedescriptions/exchange-online-service-description/exchange-online-setup-and-administration - MS official docs on setup. There are plenty of good guides on the web to check out.

Part of set up is adding/changing a DNS record for autodiscover.yourdomain.com - MS will give you all of the details you need for this, and once set up it'll automatically connect your Outlook/mobile to the mailbox without having to add specific server settings.

There is also a migration wizard in Office 365 that can help getting the PST's you have up in Office 365

Hope this helps,

Mark