is something I have often wondered about, especially since so little is written.  I am under the belief that both Tulkas and Orome did fight.  In LOTR, Tokien compares Theoden fighting in the Pelennor to Orome, who had "come long before."  The context, I think, is that of the War of Wrath.  I think the Sil also says that Tulkas fought, or I've heard that somewhere. 
    The War leaves some unanswered questions.  If the Valar still had the ability to create a beautiful land (Numenor) following the First Age, why were they not still shaping the world more generally?  Also, if Morgoth was defeated, why did the Valar retreat back to Aman?  Unless they knew from the music that there were battles with Sauron to come, why didn't they declare victory and work on healing the world?  And another thing:  why didn't they look for Sauron?  The Sil just says "but Sauron they did not find."  I find this behavior somewhat inexplicable.
   The problem of the final theft of the Silmarils is a problem.  If Maglor and Maedhros alone could creep into Fionwe's camp and slay the gaurds, it doesn't seem as though the watch upon them is very impressive. 
   As you say, we don't know very much about the army.  The Sil just says "All Beleriand was ablaze with the glory of their arms."   We do know that the Teleri, as a whole, did not participate, neither did the Elves already in Middle-Earth. 
    This post probably didn't answer any questions, but simply created more.  I do think Tulkas and Orome fought personally, and Morgoth, according to the Sil, had his feet cut off upon his capture "hewn from under him" and wore a  chain on his head "his crown they beat into a collar for his neck." My mind does not entertain the fact that:
1  The Valar didn't conduct a search for Sauron.
2 The Valar retreated back to Aman after the War of Wrath, and
3  The Valar make a perfect Numenor but don't heal the earth that has been destroyed.