How to define context for new (custom) properties of JSON document

I I have read json-ld spec @ http://www.w3.org/TR/json-ld/ but one thing
I'm having a hard time understanding how to define context for properties
(keys) that doesn't have a standard ontology yet defined by
schema.org/foaf/etc?

I created json-ld playground link (with example of json-ld document and
context) that should explain better what I'm trying to achieve @
http://tinyurl.com/k5wagv8

Also note in json-ld playground, the expanded/compact/flattened/etc format
doesn't display 'twitterId' property which I have assigned the value of
null in context-document. Why is this?

Is there a way to address this problem, or do I need to define my own IRI
for custom properties until standardized version becomes available and then
I gradually introduce new ones while keeping the old ones for backward
compatibility?

Also, in expanded format, I lose that the image is gravatarImage not just
any image. Since I couldn't find corresponding IRI for gravatarImage on
schema.org is there a way that I can preserve concept of gravatarImage in
the expanded payload.

Thanks.

P.S: Yes, I can model twitterId, linkedinId etc as Url but I'm just using
these as an example to represent databaseIDs which are not urls per say.