When checking the same Outlook.com account on 2 different locations using Outlook 2013 in both:

Some e-mails will show in one computer and in the same account in another computer doesn't receive those e-mails.

When I set up the account freshly, he downloads all e-mails properly from Outlook.com so number of emails in Outlook.com and Outlook 2013 are the same. After a few days some e-mails are missing in Outlook 2013 but show on Outlook.com

Seems to be some sort of server synchronization problem. I've had this posted in Outlook 2013 forums but they said it is a server problem, probably some e-mails on the server are corrupted and this causes problems with Outlook 2013.