Answers to both are no (or not necessarily) based on no relationships in the meaning.
Possessiveness means
Woh Mera Hai (He is mine)
or
Woh meri hai ( she is mine)
Simple and neutral emotion. Not good not bad.
It may mean I do not like to share him/her with anyone else.
Absolutely nothing to do with love or insecurity.
A person may not love other person but still want to have that person to himself or herself..right?
A person may be quite confident in relationship but still be possessive.
It is what you do out of, and why you are possessive** that may make it **love, jealousy or insecurity.