Can I qualify for RN based H-1 if I have only an Associate degree in nursing?
The simple answer is, no, unless you also have 6 years of professional experience. Also note, not all RN's can qualify for an H-1.
Do we need to send an LCA for H-1B1 Singapore quota?
Yes, you do. E-3 visas for Australia and H-1B1 for Singapore and Chile, all require an LCA.