In following sentence: "Он добавил 10 новых фото" - it appears that
фото is a direct object and so the accusative case (inanimate) should be used. However, my tables suggest that
-ых is the accusative, animate ending.
The phrase “новых фото” is in the genitive case plural.
The whole sentence Он добавил 10 новых фото is grammatically correct, it is exactly how that should be said in Russian. But the sentence is tricky for a learner of Russian since it has an indeclinable noun фото and a numeral written in digits. Moreover, the numeral 10 десять has its Accusative = Nominative, so it is not quite clear which word is in the Accusative case. But if we substitute those words for their analogs with all case forms distinct, десять for тысяча (feminine) and the truncated фото for its full form фотография, also a feminine noun with a full paradigm, we get
Он добавил тысячу новых фотографий.
Now we can clearly see that it is тысячу which is in the Accusative case, and фотографий is in the Genitive case. But why is it so?
It is all about how Russian numerals govern the case of the nouns that follow them. It is explained in this answered question: Numerals changing the case of adjectives and nouns. What is important for your sentence is that 10 десять puts the following noun phrase into the Genitive case plural:
десять чёрных котов (ten black cats)
десять спелых яблок (ten ripe apples)
десять милых женщин (ten nice women)
It is valid not only for 10, but also for all the numerals from 5 to 20 and for all the bigger numbers ending in 5 – 9, and 0. If these numerals are in the Nominative or Accusative case, they turn the following noun phrase into the Genitive case. But if these numerals are in any other case, the following noun phrase is also in that same case:
Мы подарили цветы милым женщинам. (Dative, “We gave flowers to nice women”)
Мы подарили цветы десяти милым женщинам. (Dative, “We gave flowers to ten nice women”)
Мы поговорили с милыми женщинами. (Instrumental, “We talked with nice women”)
Мы поговорили с десятью милыми женщинами. (Instrumental, “We talked with ten nice women”)
Naturally, фото is inanimate, the thing to remember is that a noun in the Genitive case can be simply a noun in the Genitive case, not in the Accusative = Genitive case of animate nouns. Special attention is to be paid to the combination numeral + adjective + noun, because it is here that all the complexity of the Russian case system and agreement is at its best (or worst).