Very thoughtful outlook on it. However, when you absorb a dragons soul you're taking his essence and knowledge to unlock power that he once had. Meaning you're taking away that very part of him which keeps him immortal and therefore once you have it he is no longer, who know's where they go but I always theorized they cease to exist. Once you have that power you use his soul to unlock whatever knowledge you wish making the soul non-existent.
And for your second question, of course not! Souls are a very dangerous thing to mess with and is a main reason necromancers have been called evil for centuries. Trapping a soul or taking a soul is just seen as barbaric and evil, your soul is needed to keep your physical form intact and alive in Mundus. Once that connection is severed you will no longer be bound to the mortal plane and your physical form will simply...die. Even the Dragonborn's have limits my dear friend, hope this helps.
You could not have spent your soul as I am sure it would not be an option seeing as your fate could not be changed no matter what. And the Dragonborn is a special case, there have been some powerful wizards in the past who tried to replace their souls or incase them somewhere to stop the fate they all faced, mortality. However, the Dragonborn faces something different as they have the soul of a dragon. Giving them to power to shout and the knowledge of the thu'um but it does not make him/her immortal as their physical form would eventually die out.
Soul Gems are crystallized magical gems that house the souls of the dead ranging from People to animals depending on the soul used. They are known to be from the Soul Cairn which is a plane of oblivion where souls captured in Black Soul Gems are sent upon use and are collected by the Ideal Masters.
So considering you were sent into Mundus by Akatosh who is a divine I highly doubt you're a Soul Gem.
No, you're Dragonborn. You're able to absorb the souls of your slain dragons and absorb their knowledge. (Only reason you can 'store' them is because the game would be unfair if you had to spend it right away) The definition of a Soul Gem is far from what the definition of a Dragonborn is.
Don't think that you can use your own soul. However, necromancer can store a soul in a phylactery to become a lich. Does it mean that Dragonborn necromancer will be able to store those dragon souls in an external location like in a soul gem? Then they might be able to use those gems to create extremely powerful artifacts. In theory
That is actually a very interesting possibility, could you imagine storing dragon souls in an external location as you said and the power these souls would grant. Grand Souls be damned, the power and knowledge stored in a dragon soul...By Azura the possibilities. In theory of course