Does Lazy Writer spend resources on flushing out pages from the buffer pool?
does Lazy Writer spend resources to make pages out of the buffer pool after periodically checking free buffer list and marking some clean pages as free


Yes it does, this is primary responsibility of Lazy writer. Its job is to make sure there is some free buffer so that new pages can be brought into memory when required.

Lazy writer would periodically scan buffer pool, look for pages which are least recently used, or you can say "cold pages" and flush them to disk. In case there is sufficient memory available Lazy writer would not do anything. In case of memory being required to bring more page into buffer pool Lazy writer would flush old pages, create space and bring new pages. It does not marks pages as clean