What is the meaning of grim-death, doughty-in-battle, fate-cursed, & battle-grim based on how they are used in the excerpt of Beowulf?

While you don't mention which translation you are using, these phrases are all what are called kennings. A kenning is a two-word adjective prominent in Anglo-Saxon poetry. They give a distinct terseness to the poetic line and conjure a powerful image in the readers' mind. These do the same thing, all enhancing the sense of the intensity of battle.
The poem focuses on Beowulf's destiny as a warrior who serves as a saving force for Hrothgar's court and then, later, the Geats. He is destined to fight monsters like Grendel, Grendel's mother, and the dragon, who are "fate-cursed." Beowulf is fated to defeat Grendel, though he is himself fated to die in battle with the dragon. A destiny shapes these warriors' lives and cultures, and they must accept their fates with courage and loyalty to king or forever be labeled inglorious.
"Doughty" is a word that means fearless, and knowing the reality (or grimness) of death and its finality requires this type of courage and resolution. Rather than enter battle in a spirit of adventure and longing for fame, the poem's warriors grimly encounter their foes and their potential death, knowing that failure to do so is failure to uphold the foundations of the community, which is based on the comitatus, or a code of loyalty, bravery, and generosity.
In the end, the Geat warrior's unwillingness to face the dragon leaves them leaderless and subject to attack by other tribes, who will no longer fear the Geats.