Jay Whitehead's database contains this game here, and this means that he had a contemporary source. Or did I misunderstand the way he worked?

As for 10...Qxe4 i dont know. I have a position in my head so say 10...Qd8 then white has 11.e5 and if say 11...Nc6 12.exf6 gxf6 13.Qe2+ Nce7 14.Ne5 h5 15.Ng6 Rh7 16.Rxf6 and i think its mate next move.

11...c6? you are right about that one. Better is probably 11...Qg6

As for 12.Nc7+ you are probably right but i think point was play for a mate not a material.

When I did my data-entry for Jay it was mostly from photocopies of material related to the German Pleiades he had gathered from periodicals he searched through in the White Collection. Sadly Jay didn't document the sources of the photocopied pages at all and typically I only had a page number from the photocopy as a reference point for my database entry (occasionally I would have the published periodical title across the top of a page). He had not gotten into the Morphy era with the material he had sent me. I have no idea what sources Jay used after the Pleiades era or if he started to document his sources.

The key of this game in JH's database is 09559. First I took the keys 0955X: two games Kennedy vs Staunton and seven games Medley vs Staunton, all played in 1849; then I took the next ten keys 0956X: there are again many Staunton games, but all played abroad in 1852, and there are two games of Anderssen: Anderssen vs Craske, London 1851 and Lange vs Anderssen, Berlin 1852. Since the games are distributed over so many years, I assumed that another relation brought them together. All games are at odds, partly at heavy odds. However, Staunton's Companion was already published in 1849, and Wray's books are on pawn and move and pawn and two moves, respectively.

Special searches for the other games yielded no surprises, too: many of the 1852 games are in the CPC, and hence I checked the neighboring volumes, as well as "The Chess Player" and "The British Chess Review". But the hope to be successful there was in any case small, because, if the game would have been published there, it would not have been in Lawson's collection.

Im not sure about title because it is in Czech but as my grandfather told me its something like "Less known games of Paul Morphy" (he knows Czech but not perfectly)

09559 is our target game

Illustrated London News:
1853.01.01: Whitehead games 10503 and 10504
1855.01.06: Whitehead game 09585
1856.01.05: Whitehead game 09623
1857.01.03: Whitehead games 09641 and 09642

Looks like 1852 or 1854 are the best bets. I don't have an online source for the Illustrated London News 1852 or 1854, so I'll have to make a trip to the library in the future (unless someone has a source and finds it first).

Also no record at the National Library of the Czech Republic: http://aleph.nkp.cz/F/BLE4KTJJPIJ6Y...

And a general internet search turns up nothing.

Very possible especially if it had a very small limited print run and was distributed locally to that region.

The New York Daily Graphic published it in 1885.10.04 as Steinitz-Amateur with the following heading: "A brilliant game at odds of QR. The following remarkable game, which has been erroneously ascribed to Paul Morphy, was played by Mr. Steinitz and a well known London amateur, and appeared in the London Era on June 14, 1863.

Jay Whitehead found it twice, once as a Morphy game and once as a Steinitz game: Game 02819 as Morphy-NN
Game 02845 as Steinitz-Rock

I recall reading an article where Reichhelm defended himself against plagiarism charges by Steinitz over a game and Reichhelm claimed that Morphy had played it before Steinitz (I'm trying to find that article now).

Looks like that game was wrongly assigned to Paul Morphy by Eugene Cook when he was sending Morphy games to Reichhlem.