Is there any Sharia-based prohibition against uprooting a sidr tree (jujube), given that some Muslims experience calamities and illnesses after uprooting it?

There is no Sharia prohibition against uprooting a sidr (lote) tree, unless it is within the حرم (sacred precinct) or causes harm or injustice to someone. There are indeed narrations that prohibit cutting them, but these are either weak, disconnected, specific to the sidr trees of the حرم, or are interpreted as referring to cutting them wantonly or unjustly. A number of scholars have issued fatwas permitting their cutting, citing the practice of the Salaf (early Muslims) in cutting them, and the Prophet's (peace be upon him) permission to wash the deceased with sidr leaves.