Russian nouns (words naming things, people, phenomena) have different genders assigned to them: masculine, feminine, and neuter. На́ша is used with feminine words. На́ше is used with neuter words. Наш is used with masculine words.
На́ши is used with plural nouns, regardless of their gender.
When talking about living people, the gender usually corresponds to their real gender:
- Он на́ш учи́тель 'He is our teacher',
- Она́ на́ша учи́тельница 'She is our teacher'
It doesn't work in 100% cases because masculine words can be used to talk about women. But in general, it's quite logical when talking about people.
When talking about inanimate objects, it gets illogical. Лампа 'lamp' is feminine, but торше́р 'floor-lamp' is masculine. You'll probably need to learn the gender of the noun. Often you can infer it from the form of the word (e.g. most words ending in -а in the Nominative case would be feminine, most words ending in -р would be masculine), but this doesn't work for all the words.
I can't hear the actual audio (I'm using my free-software-only notebook, so I don't have Flash installed here), but жи and ши are pronounced as жы and шы (in fact, жы and шы are never written; we write жи and ши instead).
The pronounciation of «наши» shouldn't change, but since it ends with the sound «ы», which is absent in many of the world's languages, you might hear the closest sound of your language instead. «Ы» is something in-between i and u, so you might hear either. You'll eventually learn to pick up Russian sounds, so please don't get discouraged!
Actually, this phrase may be used e.g. when you've come to visit someone, and you know they've assigned you some towels to use, but not sure where you can find them. Sounds pretty natural to me.
It is improper English because "towels" is plural and needs to be paired with "are". English speakers can butcher the language by shortening phrases in ways such as your suggestion. It's a lazy habit that I'm guilty of this myself, but "Where is our towels?" is still grammatically wrong.
In this case, part of the reason for the "incorrect" English is that the contraction for "Where are" is difficult to say in English: "Where're" = "Wherur" which when said quickly just sounds like "Where" - which ends up being a sentence without a verb: "Where our towels" (oddly appropriate as a literal translation of the Russians). No verb is worse English than a bad (contracted) verb.
While accepting the differences in the written spelling, it is a challenge to hear any differences between the spoken forms “где наше полотенце” vs «где наши полотенца» .... unless of course someone goes to the trouble of clear enunciation to stress the differences. The recording does not do that.
Whilst I agree that this phrase is used in English (I have heard it in Australia as well), it should not be accepted as a correct answer because when learning a language, we should be learning its grammar correctly as well. Since "towels" is plural, "are" must be used to be grammatically correct. Similarly, "Where you at?" is grammatically wrong, but used regularly. It is missing the verb "are" and the preposition "at" used at the end is grammatically wrong, http://goo.gl/uni9kD . I know that the grammar is not of the language being learnt, but it reflects on how well we learn Russian grammar. For someone learning English, it should not be a phrase used to teach. Especially this early in the Russian course, the main grammar rules are important.
Language evolves. Many of the inconsistencies we see in languages originated as incorrect grammar, but eventually linguists accepted it as a permanent feature of the language, and thus correct grammar.
I don't really have a strong opinion either way on the towel issue, but it's a common enough usage that I doubt that there's a good reason against using it here.
It's a similar situation with pedants (incorrectly) arguing the less vs fewer issue. http://www.merriam-webster.com/words-at-play/fewer-vs-less
Ultimately, when learning a new language, refusing colloquial phrasings should only be done when it actually demonstrates an important point in the target language. For example, I've been corrected on missing a word, even though the ultimate meaning of my translation got the job done. That's a valid reason to be strict with the rules because it's important to understand what that extra word means.
But unless you've got a situation like that, excluding phrasings that sound correct to most English speakers only hinders language learning. It doesn't help.