Many of the "Hesse-Others" as you call them
did only exist for few years. These splinter lines were founded because of lack of money.
The first Landgraf of Hessen-Darmstadt, Georg I, had two sons who got Landgravedoms of their own: Hessen-Braubach and Hessen-Butzbach.
Later they were joined by Hessen Homburg (which existed, like Hessen-Kassel, the eldest of the 4 "main-lines", until 1866 as independant states).
Hesse-Kassel produced Hessen-Philipsthal, Hessen-Philipsthal-Barchfeld, Hessen-Rumpenheim, Hessen-Eschwege etc etc.
As for Hesse-Darmstadt/Hesse and the Rhine: the Landgraf Ludwig X of Hesse Darmstadt had to join the Rhine-Alliance of Napoleon in 1806 and was created HRH Ludwig I Grand Duke of Hesse. In 1816 he was able to add the Rhine area to his Grand Duchy and so the title was extended Hesse AND THE RHINE.
So since 1816 the titles are Royal Highness for Grand Ducke/Duchess and Hereditary Grand Duke/Duchess; Grand Ducal Highness for the princes and princesses.
For the family members in England and Russia they were generally "the Darmstadts"