Sometimes, not always, I face an increase in line thickness while I am extending its length.
After completing this action it always jumps to its original line thickness but I find this very annoying. Is there something I can do to prevent this?

You can modify this behaviour using these buttons:
http://i.stack.imgur.com/St8so.jpg (not my image)

Find them in the toolbar (above the canvas) of the selector tool (F1 or S or just the topmost tool with the mouse cursor icon at the left).

You can switch to outlines only display mode (Ctrl+5 on the numpad a few times).
Preferably change the bounding box to visual one in the prefs (Shift+Ctrl+P) and maybe in the document's settings (Shift+Ctrl+D) change background colour (outline mode may look too contrasting).

If stroke width is not that important, switch to scale strokes with the object.

Assuming you want to preserve an even stroke width regardless transformations, I don't know a real solution of that behave when in object select mode.
However if you use the edit tool and select all nodes to be resized, you can skip adding a transformation to the object itself.
Which basically has the same effect.

I think AnalogueMan is looking for a workaround to avoid a known issue (a temporary effect during transformations (e.g. scaling) of stroked objects with the mouse):

• Bug #165727 (sf969406) “display is wrong when transforming by mouse”

[ I'm not aware of an existing workaround to avoid this temporary scaling of the stroke width during transformations apart from: don't scale a line with the select tool if you want to change e.g. the length of a straight horizontal or vertical line - use the node tool to edit the geometry. The select tool will transform the whole selection based on its bounding box (defaults to visual, i.e. includes the stroke width, markers, the filter effects region, etc.), and does not directly edit an individual geometric parameter of the object. ]

Ah, yes, I didn't read thoroughly... sorry!

Thanks you forum members for your responses. I think I will use the suv solution as the most convenient one: use the node tool to edit the geometry.
Reading the Bug report, I noticed that this was reported quite a long time ago........