What options do I have when creating menu with options (contextual menus). What is allowed and what does Apple provide?
I have a toolbar with buttons and a click of a button opens a menu with options:
Examples:
UIPopoverControllers are only available on iPads, but it's ok to create similar looking things for iPhone
What you see in the picture, is entirely custom. a UIView subclass likely. There are pretty much no limits to what you do with your UI, within reason, so no ugly highly contrasting primary/secondary colours, and it's best to keep it at least similar to the native UI. If you say use a windows 7 phone ui style in an iPhone app it's more likely to get rejected by Apple, than keeping to the style of iOS.
So make it look as "professional" as possible, using gradients and shadows, edge arrows, like in the above picture.
Apple provides pretty much nothing that your looking for,