refactor: remove dead code, DRY calendar navigation and helpers

- Fix Jalali calendar weekday offset in buildCalendarMonth
- Remove unused gregorianDateKey function and PairCount field
- Replace inline export month title with formatMonthTitle
- Extract navMonth helper for prev/next month navigation
- Extract sendOrEdit helper for send-vs-edit message pattern
- Extract monthGregorianRange helper to simplify editCalendar 3-way switch
- Replace export displayDate conversion with formatDateForCalendar

internal/bot/dateutil.go

@@ -211,9 +211,12 @@ func buildCalendarMonth(cal string, year, month int) calMonth {
 
 		gy, gm, gd := jalaliToGregorian(year, month, 1)
 		dow := dayOfWeekGregorian(gy, gm, gd)
-		_ = dow
+		startOffset := (dow + 1) % 7
 
 		var days []calDay
+		for i := 0; i < startOffset; i++ {
+			days = append(days, calDay{dayNum: 0, date: ""})
+		}
 		for d := 1; d <= totalDays; d++ {
 			gy, gm, gd = jalaliToGregorian(year, month, d)
 			days = append(days, calDay{
@@ -267,10 +270,6 @@ func buildCalendarMonth(cal string, year, month int) calMonth {
 	return calMonth{title: title, weekDays: weekDays, days: days}
 }
 
-func gregorianDateKey(year, month, day int) string {
-	return fmt.Sprintf("%04d-%02d-%02d", year, month, day)
-}
-
 func parseGregorianDateKey(key string) (int, int, int) {
 	var y, m, d int
 	fmt.Sscanf(key, "%04d-%02d-%02d", &y, &m, &d)
@@ -290,3 +289,27 @@ func formatDateForCalendar(date, calType string) string {
 		return date
 	}
 }
+
+// monthGregorianRange returns the Gregorian start and end date strings
+// for a given calendar month (cal=gregorian|jalali|hijri).
+func monthGregorianRange(cal string, year, month int) (start, end string) {
+	switch cal {
+	case "jalali":
+		gy, gm, gd := jalaliToGregorian(year, month, 1)
+		start = fmt.Sprintf("%04d-%02d-%02d", gy, gm, gd)
+		lastDay := monthLenJalali(year, month)
+		gy, gm, gd = jalaliToGregorian(year, month, lastDay)
+		end = fmt.Sprintf("%04d-%02d-%02d", gy, gm, gd)
+	case "hijri":
+		gy, gm, gd := hijriToGregorian(year, month, 1)
+		start = fmt.Sprintf("%04d-%02d-%02d", gy, gm, gd)
+		lastDay := monthLenHijri(year, month)
+		gy, gm, gd = hijriToGregorian(year, month, lastDay)
+		end = fmt.Sprintf("%04d-%02d-%02d", gy, gm, gd)
+	default:
+		start = fmt.Sprintf("%04d-%02d-%02d", year, month, 1)
+		lastDay := monthLenGreg(year, month)
+		end = fmt.Sprintf("%04d-%02d-%02d", year, month, lastDay)
+	}
+	return
+}