Db Piccolo's first came about when bands began to be dominated by early brass instruments, and repertoires were adjusted to suitable keys for brass instruments - which created awkward fingerings for the old, simple system, C piccolos. (The simple system limited the keys the instrument could easily play in.) By creating the Db piccolo, piccolo players were instead able to play in the same key's as they had previously been doing in orchestras, making parts more playable.
In later years, Boehm system instruments were created, which allowed all keys to be easily playable. At this point, the Db piccolo began to die out, as players could own and use a single pic for all pieces/ensembles - the C piccolo was pitched as the flutes, also being the obvious logistical choice there. [Source]
An alternative explanation, offered by Kim J. Teal is that the tone of the C piccolo was superior - however, I think the first explanation sounds likely to have been one of the heaviest contributors.
Today, I'm lead to believe the Db piccolo has been replaced by the C pic as a standard, worldwide. (Mainly based on the apparent lack on manufacture.)
To me, it's strange that the Eb Horn was included alongside the Db Picc here, in the UK, I would suggest Eb Horns are potentially more common than French Horn's, due to their use in Brass Bands. In the UK, they're referred to as Tenor Horns, which I've just learnt relates to the Alto Horn, in the US.
The Eb Horn was developed by Adolph Sax (also the maker of the clarinet, if memory serves), in 1840s. [Source] As such, I would assume that the modern US 'Alto Horn' is the Eb Horn many pieces refer to - as it seems an appropriate time pieces. (I'd be interested to hear if you still have reason to believe not, OP.)
As to why the French Horn is more common in instrumentation, I would imagine this would simply be due to being the more established instrument, and its common orchestral use. In the UK however, many wind band arrangements come with horn parts in both Eb and F (the same parts transposed) - so that Eb Horn players don't need to transpose on the fly.