Leistungsfähiger Amerikan (were you someone else once?) you definitely raise some good points
Having the civil war break out around 1858 should allow the South to retain a degree of balance against the Union; whilst increased militarisation will certainly have occurred it will be somewhat more evenly spread, although the Union fleet is likely to be larger, not least because when the civil strife breaks out, the Union is the commander-in-chief of mobile forces and can order them home to loyal ports.
British naval dominance may well not look all that overwhelming at this period, because even historically France had a pretty strong fleet. Here, the Orleanists remain, there is no break in naval development, and the arms race with the newly-unified Germany has focused their minds. Having said that, the Anglo-American War has likely made of the British fleet a more mobile and veteran force, and given a qualitative advantage to the British.
The French and the Spanish, though, would be allied with the Union, and in combination would pose a definite threat to British naval dominance. That said, unified Germany has a fleet of some standing and France will have to retain a strong enough force in European waters to counter and potential threat.
I think the idea of Russia intervening on the Union's side here makes a good deal of sense. There has been no Crimean War, for one thing, and Franco-Russian relations are not at any real low ebb (not least because in an Orleanist king, the Tsar sees an equal of ancient bloodline rather than as in OTL a parvenue Bonaparte). A Franco-Spanish-Russian alliance in Europe, facing off against an Anglo-German one would certainly be interesting !
Whether the Ottomans, or Egypt, come into this is questionable. Russia's immediate ambitions in the Principalities could well already have been achieved short of war, and the Holy Places row over Jerusalem is likely not to rise so high in the ATL. However, for every conflict averted and every argument that doesn't happen, the law of averages usually state that something else does instead. But these do not need to include the same protagonists, involved in the same way.
If Russia has taken greater advantage of the collective post-1848 headache, then the Ottomans may very well be in the Russian camp (as per Unkiar Skelessi) and unmolested by their neighbours. Egypt is in transition, after the deaths of Mohammed Ali and Izmail and a period of weaker rule. Said is coming into his own, but its some way back and whilst France and Britain both have interests in Egypt, perhaps it is to a united Germany that Egypt looks, whilst covering its bases with Russia.
It would thus be very much in both Istanbul's, and Cairo's interests to remain neutral, and certainly in the first stages of the war I don't see anything to challenge this. Russia has Germany to fight, and Austria to look at.
Austria, for its part, is weakened over OTL by having seen a united Germany emerge from 1848, and an independent Milan under French protection. However, Milan actually provides in a curious irony a barrier to Piedmont-Sardinia's irredentist ambitions, and makes Venice safer under Austrian auspices. Of course, this advantage over OTL won't be at all obvious in Vienna, but does exist. In another curious irony, it is to both Austria's and France's advantage to see Italy remain divided and not unify - Austria because it protects their holdings, and their client states, France because it protects their direct say in the peninsular, and how this is manifested in Milan and in Rome, with the Papacy.
However, whilst sharing common goals, they have a definite rivalry for the play of the power they both wish to see exercisable within Italy, and are hardly likely to come together due to this. Whilst many in Vienna also smart over exclusion from the rest of Germany, there are also many who see the final line of that story as having now been written and that it would be foolish to try to force a revision when its all over and done with.
I certainly see Austria's position at this stage of the conflict as being one where they wish to avoid any entanglements, and where they will look to their own interests, and protect these, rather than seek to involve themselves in external conflict.
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Theatres
In a war with the sides outlined above there would be several immediately identifiable theatres
In N America
-- 1. Oregon borders, along the Colombia River and pretty much bounded by the Rockies
-- 2. California, where Mexico can launch assaults from the South, and Britain by navy from North or East
-- 3. The Mid-West, where both British and American forces are going to be light, but with Indian allies, the Great Lakes, and some important trading cities there will certainly be campaigns
-- 4. New England, where the main British offensive/defensive effort is going to be concentrated, and where the US also is going to be able to deploy forces in number
-- 5. The Great Plains, where CSA and USA forces, light in number and with Indian allies, are going to clash in their Western borderlands
-- 6. The Ohio, where the trans-Appalachian Southern states and the Old North-West of the Union are going to clash across and upon the rivers and where a lucky or particularly skilful general could turn the flank of their opponent and inflict a decisive defeat
-- 7. Mason-Dixon, where the main US and CS armies are going to be manoevring and clashing, where naval support from their fleets and those of their allies could well prove decisive in localised conflicts, and where a battle for blockade supremacy is going to be fought
-- 8. The Caribbean etc, where the combined fleets of the CSA (such as it is), Mexico and Britain will initially hold sway, but where naval task forces from France, Spain and the USA will be attempting to challenge this supremacy
In Europe
-- 9. The Anglo-French sea border, also including the civil war in Ireland, as well as Spanish and potential Russian involvement in this theatre
-- 10. The Franco-German border, also including the possibility of it spilling into Belgium or Holland-Luxembourg
-- 11. The Russo-German border where the main Russian effort is going to be concentrated, and where both nations will fight each other navally in the Baltic for control of the sea
And considering
-- 12. The Baltic, White Sea etc where British naval taskforces may well be sent to challenge Russian forces, Russian control
-- 13. The Mediterranean, where British naval taskforces will be challenging the Spanish and French, and probably also Russian forces to be found there
-- 14. The N Atlantic, where both alliances in full will engage in commerce raiding, trade protection and the hunting down of each other's forces
-- 15. S Atlantic, Indian Ocean, Pacific etc, where naval forces from the maritime powers will engage in colonial raiding, small-scale landings, support for native rebellions, and commerce interception
-- 16. The Far East, where Britain, increased in strength after the Anglo-American War, will be able to take the war to Russia with raids into the Sea of Okhotsk, and potentially involve China and Japan in goings-on in this theatre
Best Regards
Grey Wolf