"The box was initially moving." is a perfectly fine sentence grammatically, but it might not make much sense in most contexts. Like Jim mentioned in the comments, this would probably be most likely seen in a physics word problem.
I'm not an expert, but I do believe the last two sentences are also fine. I don't know if you're looking at a multiple choice question which forces you to pick between those two sentences, but if I had to guess, I would pick the first one. My reasoning is that although the second sentence looks correct, it's somewhat unusual to replace an infinitive with "as." I do think the second sentence is correct, mind you, I'm just not sure if it's a forced choice (since you capitalized and italicized the "or"). I believe it's being used as a preposition here, and it's in accordance with this definition:
- In a manner similar to; the same as: On this issue they thought as one.
My reasoning here being that the structure of this example sentence and your last two sentences is the same around the word "as."
Source: The American Heritage® Dictionary of the English Language, Fourth Edition copyright ©2000 by Houghton Mifflin Company. Updated in 2009. Published by Houghton Mifflin Company. All rights reserved.