What you can do, if not all parts of the lines shows up, is to open the chart properties, go to the "General" tab, switch to "Bar Chart", go to the "Presentation" tab, select "Allow Thin Bars" and "Show All Bars" and finally go back to the "General" tab and switch back to Line chart. Then the full line should show up.
By the way - you can change the number of columns and rows of charts in the Trellis chart, You'll find the settings in the Dimension tab under the button "Trellis..."
The problem is that the graphs are updated with new dates every day and after some time there will be so many dates on the x-axis that you will not be able separate dates from each other. (not sure how Qlik handles this?)
With the x-axis scrollbar you could limit the number of dates seen in the graph to e.g. 10.
Maybe this is not possible when using Trellis graphs.
A workaround: You can always limit the selection instead. See attached where if you have selected "Date" in the dimension group, you will see the last 10 days. If you select a date you will see the last 10 days up to that date.