In English, there are several good reasons (based to think of determiners as constituting a part of speech of their own separate from adjectives.
(1) Adjectives can be suffixed with -er or -est, but determiners can't. ( taller/biggest vs. *many-er / *the-est ).
(2) Adjectives can be intensified (with words like very or really ), but determiners can't. ( very weak person vs. *very her book )
If we define a part of speech as a group of words that pattern together in terms of structural criteria like these, then observations like (1) and (2) compel us to conclude that determiners and adjectives are indeed separate part-of-speech categories.
For further reading, a quick Google search on the topic turned up some good discussion on the topic at partofspeech.org and yourdictionary.com.
As for your second question ("whether interjections are a part of speech"), it is more of a philosophical question. On one hand, by definition, they aren't grammatically related to the rest of the sentence, hence you can't use syntactic tests like the ones I used above. On the one hand, though, this very fact can be used as a defining criterion for them. (In other words, it is the only 'part of speech' that is syntactically divorced from the rest of the sentence.)
I think most linguists today would acknowledge interjections as forming a distinct class of words. Lists of interjections can readily be found online (e.g. YourDictionary.com, EnchantedLearning.com, VidarHolen.net, DailyWritingTips.com, etc. ). However, if you compare several such lists, you will find that there are certain ones that all lists have (the more 'prototypical' ones) and others are more idiosynctatic and only found on 1 or 2 lists. Due to their nonsyntactic nature, there is certainly something paralinguistic about interjections that make them more of an open-set class of words than other parts of speech. (For example, if you hit your toe on a desk, you could yell out an interjection that is not an established word of English but nonetheless communicates your pain.)