If mentally opening the box and checking to see if the items are there yet makes you feel a sense of anticipation and excitement knowing the items will eventually arrive, then it will assist the manifestation process.

If mentally opening the box and checking to see if the items are there yet makes you feel a sense of lack and disappointment because the items haven't arrived yet, then it will inhibit the manifestation process.

I agree with Vesuvius's answer regarding your first question.

Regarding your second question...

Can one state the desire to get rid of a current limiting belief and put that in the box?

...yes, of course you can - you can request anything you like.

With requests like that one, the answer may come in the form of some method or process that you will feel inspired to try out (this has happened to me many times) or perhaps you will gain some insights into what the limiting belief is all about...which again is something that has happened to me on a number of occasions.

Personally, I think limiting beliefs are basically at the root of every problem in life.

I have yet to come across any life issue that could not be resolved by expanding your existing belief about it and allowing a more helpful manifestation to come about as a result.

So, for that reason, the study of beliefs and belief-change is probably my most passionate hobby because it is so fundamental to just about everything.

And I think all that came about for me as a result of consciously deciding at some point that I wanted to get rid of a few of those limiting beliefs...in other words, I made a request. :)