Prepositions within multi-word conjunctions
Q1. ___ the rain, we went out
IN SPITE OF is correct. DESPITE alone is also correct. Rain did not deter.
Q2. ___ you arrive, call me
ON (meaning when) introduces the condition. Arrival prompts calls. Communication is maintained.
Q3. ___ studying, he works part-time
WHILE requires a clause or gerund. Work and study coexist. Time management is tested.
Q4. ___ his illness, he attended
NOTWITHSTANDING is highly formal. Attendance occurred. Illness was overcome.
Q5. ___ the weather, the event proceeds
REGARDLESS OF means without consideration. Events continue. Weather is irrelevant.