Try reporting it. Wiktionary does list it that way, but I have seen and heard it used as a more general term that includes time before and after noon as long as it is close to noon. http://www.merriam-webster.com/dictionary/midday
12:00pm, noon is not "at some point" after morning but before evening; it is a very specific minute of the day.
You can say this, but it assumes a very special context. If the other party is full of doubts about the drivers' trustworthiness and alike, and brings new arguments against your travelling plan, you may close the argument with this, like "At noon, the bus will be here [and there is no other option]" = "Délben a busz itt van [és nincs másik lehetőség]". (Don't detain me, mate, I won't change my mind!) ;)
In normal daily usage this word order is extremely rare.
It is idiomatic I think. (And this is "reggel" with single L and double G—just for reference :) ) It is the same with evening. Evening = este In the evening = este night = éj, éjszaka, éjjel in the night = éjjel, éjszaka.
You may notice a scheme with "reggel" and "éjjel" as "éj" is a bit archaic/poetic. "Reggel" has an exinct/poetic form of "reg". Both expressions can be understood as "with morning" - reg+vel --> assimilating to "reggel"; "with night" - éj + vel --> assimilating to "éjjel". As if the time would accompany you. ;)
But I am not sure about this etymology, as this is more like an amateur's deduction.
No, there is no such word. Technically it could exist as "reggelben" but it is completely different in its meaning. A rare example when you can use this is "I hate in Monday mornings that I must go to work" = "Azt utálom a hétfő reggelben, hogy dolgozni kell menni". But please be aware the big difference that here it is in singular in Hungarian. When you speak in general, you can often use singular forms, but it has the same meaning when this is plural like "Azt utálom a hétfő reggelekben, hogy dolgozni kell menni". The "-ban, -ben" refers to time only for "dél" (noon).
Also pay lot more attention to spelling, because Hungarian words can mean awkwardly different thing when misspelled. Even the accents are critical. There are several proverbial examples like "csíkos" vs "csikós", "kérek" - "kerek" - "kerék", but also such "rezel" vs "reszel", "var" vs "varr", "arra" vs "ara", etc...
It's the same in many languages, even if native speakers aren't aware of it. E.g. there is a big difference in English between "morning" and "mourning" or between "son of a bitch" or "sun of the beach" ;) For a foreign speaker, especially for a beginner, they sound desperately similar, but not for the English speakers! ;)
Ages ago there was a film with Sylvester Stallone (Over the Top) where he tells about a dream to his son to make a venture with him and the Hungarian voice-over did not get the hang of it: the original "Hawks and Son" was translated to "Sólyom a napban" ("Falcon in the Sun"). Such errors were pretty common in the 80s :D