There's a difference between written and spoken French.
Ideally, in written French you would never drop any of those, unless you're talking in a really unformal context, like students or friends conversations. But even with friends, if you can avoid dropping words in writing, it won't hurt you and will definitely not make you seem too formal.
In spoken French, it gets tricky. You can pretty much drop the "ne" in most cases, even at work. I fail to find a sentence where dropping the "ne" would be an issue. If you need to show a very high amount of respect for the person you're talking to, don't drop it. Otherwise, it doesn't really matter.
que, however, is not entirely dropped in most cases. In your example, as people said in the comments, the
que is not dropped.
ainsi que ma trousse becomes
ainsi qu'ma trousse and the
qu is almost silent. The two syllables of
que ma merge into one syllable, which is why you can't hear the
qu unless your ear is really trained for it.
ainsi ma trousse doesn't make sense in French in that context, and that's why there is absolutely no doubt that the
qu is pronounced here.
As opposed to the
ne, there are very few instances where you can really drop the
que in spoken French. One example that comes to mind would be
qu'est-ce que becoming
qu'est-ce. For example, in spoken French, it's not that rare hearing someone say
qu'est-ce t'en penses instead of
qu'est-ce que tu en penses.
That being said, if you don't drop the
ne and you don't drop the
que, you will not sound weird in most cases. People will notice you're not a native speaker anyway, so my advice, even though not asked for, would be to keep the
ne and keep the
que. That will help you with written French and you can always shorten things up later based on your conversations with native speakers.