British, with their successive naval dominance, first following the battle of the nile and then the battle of Trafalgar, it strikes me odd that - as far as i know - the british never threatened an invasion directly into france while Napoleon was busy in either Austria, russia or spain.
British campaigned instead in spain and then later in france. But couldnt even simple raids to north of france and/or threaten a move on paris been possible? Were any of the sort going on?
Were the central of france heavily garrisoned etc? Werent the british naval superiority able to support an expidition?
Under Napoleon, the French military was arguably the best in the world, having bested the armies of Austria, Russia, and Prussia. Like you yourself said, at that time the British were traditionally known to be a naval power. Though their army may have been as professional and disciplined as that of the French, it was always vastly outnumbered, so risking a potential run-in with French troops was out of the question. Direct British support in the Peninsular War happened because there was a potential for asymmetric warfare against the French due to the harsh terrain and local resistance.
As to the second part of your question, expeditions into France itself weren't possible because in the early 1800s, Napoleon had stationed most of his army in the north of France, prepping them for his planned invasion of England, which was only scrapped due to the outbreak of the War of the Third Coalition on the European mainland.
So basically it was impossible because they simply lacked the numbers. Even when Napoleon was off fighting somewhere else, the French military always had a steady flow of new men which could easily field a new army in defense of the homeland.
You may wish to look up the Walcheren Expedition. It is the 1809 invasion of the Netherlands in order to open up another front, and whilst not France proper, is still close enough to be considered a threat perhaps. What is important is that it was absolutely smashed, and probably from that point onwards discouraged any similar such ventures.