The "fourth sword" is actually an upgrade to the Master Sword, which is the third and final sword you obtain. The upgrade (earned automatically in the Palace of Twilight) allows you to repel the purple fog, kill many twilight monsters with one hit, and light certain sacred orbs.
The Master Sword.
You only have it in the twilight world so whenever you are in the twilight world you have the light sword.
no
The Sacred Grove
In my opinion skyward sword and twilight princess are very alike! Twilight princess is a bit longer though. (I recommend Twilight Princess because the plot is more interesting)
Yes
If by "second Twilight Princess" you mean Skyward Sword, no, it has not, or I would have already bought it. Also, Skyward Sword (in the chronological timeline order) occurs before Ocarina of Time and therefore before Twilight Princess as well, meaning that it is not a sequel, but a prequel.
Your sword cannot burn out. Only the shield can and it can be bought in most shops
Two: Skyward Sword and Twilight Princess.
Get next to it and pick it up.
There isn't a sequel YES. it is skyward sword
After getting the master sword.