Word and Writer are based on different principles resulting in different behaviours for seemingly identical specifications. I assume that your document is presently .doc(x) which means it must undergo a conversion when read into Writer and yet another (not exactly inverse) conversion when saved. This adds the complexity of compatibility to your learning curve.
To help you efficiently, mention OS name (presumably W$, but which version), exact LO release (with the four components in the number) and save fornat (again, presumably .doc(x)).
What do you mean? When you add text in a cell, making this cell taller, subsequent rows are automatically shifted to make room for the larger cell.
Perhaps the properties of your table don’t allow it to be split between pages, or its rows. In this case, page breaks occur exclusively between rows, not in the middle of a row.
Are you trying to drag’n’drop some data?
Since a table is a “complex” object, I don’t trust drag’n’drop here. I prefer traditional copy-then-paste as it allows a finer control on the operation.
Is it made of many rows? Or a few “giant” rows?
What is in this table? Is the data really tabular “by essence”? Monster tables (those with cells spanning several pages) put a considerable stress on Writer because of non-trivial layout issues.