You cannot omit δικό in a phrase with this structure. You can use μας on its own if it is to say 'our' and it goes straight after the noun: our cake = το κέικ μας. To say 'ours' or 'our own' both δικό and μας (in whichever gender, case and number the noun is) are required.
No, since the accent is on the ε it's clear that it's pronounced as a separate letter and not as the diphthong..ει as in "είσαι so the diaeresis (διαλυτικά) are not used...although they may be seen in older uses.
Google "κέικ" and "τούρτα" and check out the images.