No, they don't have a shared root.
Of bobbin (in a sewing machine), the Online Etymology Dictionary says:
1520s, from Fr. bobine, small instrument used in sewing or tapestry-making, perhaps from L. balbus (see babble) for the stuttering, stammering noise it made.
Of bobby [pin] (a woman's hair pin), the Online Etymology Dictionary says:
1928, from dim. of bob (2) + pin (n.).
Where bob (2) is:
"short hair," 1680s, attested 1570s in sense of "a horse's tail cut short," from earlier bobbe "cluster" (as of leaves), mid-14c., a northern word, perhaps of Celtic origin (cf. Ir. baban "tassel, cluster," Gael. babag). Used over the years in various senses connected by the notion of "round, hanging mass," e.g. the meaning "weight at the end of a line" (1650s). The hair sense was revived with a shift in women's styles early 20c. (verb 1918, noun 1920). Related words include bobby pin, bobby sox, bobsled, bobcat.
Neither bobech or bobeche are in the Online Etymology Dictionary or Oxford English Dictionary, but Wordnik says bobeche (with an -e) (a glass collar on a candle) is from the French bobèche.