In general, no implications hold (in either direction) between the Lindelöf notion and other compactness notions, such as paracompact, which are discussed in the compactness page. Any second countable space is Lindelöf, but not conversely.
However, the matter is simpler for metric spaces. A metric space is Lindelöf if and only if it is separable if and only if it is second countable.
An open subspace of a Lindelöf space is not necessarily Lindelöf. However, a closed subspace must be Lindelöf.
Lindelöf is preserved by continuous maps. However, it is not necessarily preserved by products, not even by finite products.