Not trying to be a dick - but when a guitar is that cheap, there are often major issues associated with it. Everything is slapped together and quickly/cheaply made - and unfortunately, the neck and frets (the most precision parts of the instrument) often have problems.
Not a chance... All a string tree does is increase pressure over the nut. It has no structural importance whatsoever.
The flip side is that string trees (so I've read) tend to not be as nice for string bending, though maybe that depends on if you get a crappy string tree, or nicer ones that don't cause friction on the string.
There's a problem for every solution.....behold, roller string trees!