I moved house recently too, (in my homeland, from one city to another) I had too many little items, not very valuable but still costs remarkable amount of money to buy. I don't know how many items are missing yet. I could only account for rather expensive ones and big size electronics like tv, rest I would only know when I'd need that particular item.
Also covering with plastic film wasn't done nicely, my furniture got disturbing scratches. To save space, they fill my wardrobe with boxes of my belongings, as a result back of it and the doors gave way
This was first time moving house in my life, before settling in Singapore I wanted to give my parents my rather new furnitures and items, instead of selling them off cheap, I paid money to move them...
Now I have some guidelines of what to do from my experience:
Start packing everything 2 or 3 weeks before you move, if possible, start with small things that you might not need anytime soon, like tools, some accessories, extra cables, some of kitchenware.
Box everything yourself, and tape the box with paper tape and mark it with X using a large point marker or stamp if you have, at the edge between paper tape and the box itself. This will help you see if boxes were ever opened.
Don't let them store things in your storage furnitures, like wardrobes. They move freely during transport and when there is a movement they will damage the furniture.
Buy a very big roll of stretch film, can be the kitchen version too. Wrap furnitures more than 5 times at least, though this will not protect against sharp items or negligent movers, it will at least minimize the damage.
This way you can ask for discount too, since they will only carry, won't do any packing.
