Because seventh chords have four notes, wouldn't it make them have more inversions than a triad?
Yes. They have root position, first inversion and second inversion which we have also seen with triads and are exactly the same with the root on the bottom, the 3rd on the bottom, and the 5th on the bottom respectively. However we also now have third inversion which has the 7th on the bottom.
yes, they do. You are correct.
Yes - the more notes avaiable in any chord the more inversions you can have.
An inversion is named purely by its lowest note, so since seventh chords by definition are comprised of four notes, ther will be, in rising order named from the lowest note they are roo, 1st, 2nd and 3rd inversions. The last probably sounding the weakest.
Of course with four notes, there are more voicings available as well.
Root in key C could be C E G B, C E B G, C G B E, C G E B, C B E G, or C B G E. All of which are playable on piano at least.