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LONDON MONARCHS F.C.
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BIRMINGHAM STEELERS F.C.
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BERLIN CAPITALS F.C.
The Brandenburg Gate features at the top. The black bear is from the flag of Berlin and also the Coat of Arms, as is the overall colour scheme.
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MADRID TOROS C.F.
C.F. is for Club de Fútbol, as per Real Madrid C.F. and as with that club, I have used the Spanish Crown at the top of this badge.
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LAZIO S.C.
The Italian flag and the SPQR seal of the Roman Republic, featuring the Roman Eagle, reference the Regione Lazio Coat of Arms, as requested. For yellow, I have used the gold in the edging and the lighter blue is used in a blue-and-white stripes combination, similar to the Argentina national team.
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PARIS REVOLUTION F.C.
There is a white outline on the outer edge of this logo.
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F.C. HAMBURG
There is a white outline on the outer edge of this logo.
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CATALANS C.F.
I created this logo a while ago for somebody else but it's very similar to what you're requesting so I trust it will serve the purpose. The Coat of Arms and flag of Catalonia are featured in the bottom section. The light-blue-and-white at the top is a Lionel Messi-inspired reference.
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MILANO F.C.
Combining elements from the Coat of Arms and flag of Lombardy, the region in which Milan is located.
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AMSTERDAM DISTRICT F.C.
An arched brick bridge, such as those you'd find straddling the canals in Amsterdam, features at the top of the shield. The flower is a tulip. The black-and-red stripes with the three white crosses running down the middle is based on the city's flag and Coat of Arms. There is a white outline on the outer edge of this logo.
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FV BAVARIA MÜNCHEN
Using the Barcelona template, the Rampant Lion of Munich appears in the top left section. The Alps are represented top right. The flag of Bavaria fills out the lower section. The German word for "Football Club" is "Fußballverein", which means that your FC Bavaria becomes FV Bavaria — consistent with the German word "München".
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ÖSTERREICHS FV
I have Germanicised your Austria FC, on the basis that football clubs are usually created in their own language — even if that name is translated by foreigners when they refer to the club. For example, the word that appears on the badge of my Ukrainian sister-in-law's team looks more like "Waxtap" to an Englishman's eye but translates as Shakhtar. The ferris wheel represents the Wiener Reisanrad. The top right section combines the Austrian Flag with the country's Coat of Arms. The section at the bottom is the Flag of Vienna, which — given the location of the Reisenrad — is where I interpret this club to be from.
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STOCKHOLM VIKINGS S.C.
Stockholm is both the capital city of Sweden and the name of the county, so there are two Flags of Stockholm; the outer roundel uses the colours of the City flag and the blue, yellow and red in the middle is taken from the County flag. Vikings did not have horns on their helmets — this particular vector image of a viking did have horns but I have knocked them off in the interests of authenticity (...and because a previous requestor asked me to). According to the online translator I use, "Stockholm Vikings" translates as "Stockholms Vikingar" but if you add "Sports Club", it reverts to the English term.
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MARSEILLE DOCKERS C.F.
The shape of this badge and that light blue cross with the black outline are lifted from the centre of the Marseille Coat of Arms. A cargo ship, anchors and a football identify this as the club of the stevedores (or dockworkers) of Marseille.
That's your lot. With the possible exception of the Birmingham Steelers one, they are not really in the MLS style but I think you overestimated my design ability with that request — I just try to come up with an idea and action it as well as I can. If I manage to create a logo in a particular desired style, then it's kind of coincidental!