There’s a difference between written and spoken French.

Written 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.

Spoken French

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.

The 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.