Ok, if you say "there is a dragon with me" - "Mae gen i ddraig", why, if you negate, you do not use the normal negation form "Dydy gen i ddim ddraig"
Both does and dydy are "normal" :)
As I understand it, oes? and does are used when "to be" talks about the existence (or non-existence) of something indefinite -- "Is there a dragon in the room? Do you have an apple? There are no apples on the table. We do not have any children."
And ydy? and dydy are used when "to be" talks about the existence (or non-existence) of something definite -- "Is your book on the table? The children are not in their room."
When you are not asking a question or negating anything, both of them are mae. (Some speakers -- perhaps under the influence of English? -- use mae 'na where English would "there is", e.g. Mae 'na llyfr ar y bwrdd "There is a book on the table", while others use simply Mae llyfr ar y bwrdd.)
So since "there is a dragon with me" is about the existence of something indefinite, the question form is oes? and the negation is does dim.