Outlook.com & Outlook for Mac

I'm a newbie when it comes to Outlook.com  -- I have a Mac with OSX Mavericks. I travel all over the place and find it hard to try and sync my mail from one device to another.  I've added my corporate email now to Outlook.com while on the road, and need to know if I send email/receive email to/from my corporate account using Outlook.com, when I go to my office, will that email magically appear in Outlook for Mac?  If so, will it replicate on a Macbook Por and Mac Mini using Outlook for Mac?  I know it sounds absurd, but I've been having to use Gotomypc to get into my email remotely, and now with the latest release of Mavericks, GOTOMYPC does not work so well.  So --need to come up with a solution. 
I currently use Mac Mail , but can't open it on two devices, as for it will not populate both systems to sync properly. 

So, I'm hoping that Outlook.com will work when i travel and get firewalled, and once back in the office, I can download everything or have everything magically appear on my Outlook for Mac.

Lastly, can you add an image in the signature line of outlook.com mail?

Appreciate any assistance on this.

Thanks,

S