How do Context Sensitive Grammar systems work?
Your understanding of Linear Bounded Automata is incorrect.  These automata are NOT derived from Pushdown Automata - there is no 'stack' in them.  A closer understanding would be that LBA are a special, 'truncated' form of Turing Machine, in which the size of the tape is restricted to being linearly-proportional to the size of the input string (actually, with a large enough alphabet, the tape space can be limited to -just- the space taken by the input string).
Also, the grammar-expansion behavior described in (1) is more closely analogous to a Lindenmyer System, which is not quite the same as a CSG (in which only one substitution takes place at any one step, so (3) isn't a concern).