Does "va'Yishkav" imply that he died on his bed?
Rashi: No. Nebuchadnetzar bound him in fetters to take him to Bavel, and they dragged him and he died [amidst this] in their hands - "Kevuras Chamor Yikaver Sachov v'Hashlech" (Yirmeyah 22:19), and in Divrei ha'Yamim II, 36:6 it says that they fettered him to take him to Bavel.
Was he buried with his fathers?
Radak (1, 6): No. It means only that he died, like "va'Ye'asef El Amav" (Bereishis 25:8 and other places). His dead body will be cast [on the ground] in the heat of day and the cold of night (Yirmeyah 36:30). 'Rashi' (Divrei ha'Yamim II, 36:6) - "Kevuras Chamor Yikaver" (Yirmeyah 22:19) - his 'burial' was like a donkey's, in a dog's stomach (Vayikra Rabah 19:6).