The cell nucleus houses the chromosomes, which are linked to a soft shell of lamin filaments. Experiments indicate that correlated chromosome dynamics and nuclear shape fluctuations arise from motor activity. To identify the physical mechanisms, we develop a model of an active, crosslinked Rouse chain bound to a polymeric shell. System-sized correlated motions occur but require both motor activity and crosslinks. Contractile motors, in particular, enhance chromosome dynamics by driving anomalous density fluctuations. Nuclear shape fluctuations depend on motor strength, crosslinking, and chromosome-lamina binding. Therefore, complex chromatin dynamics and nuclear shape emerge from a minimal, active chromosome-lamina system.