Hello Steve,
By modifying an interface cost you are basically influencing "best route" calculations.
There are two main ways to change interface costs, by modifying bandwidth and establishing a specific cost in the interface.
As a basic example, lets say Router 1 and Router 2 are both advertising route X and both are advertising this route with the same cost to Router 3 (Therefore, no best route exist, both are equally good). If you changed the cost of the interface in Router 1 to a lower cost (The lower the better) then Router 3 would receive Route X with a better metric from Router 1. Router 3 will prefer the route to reach Route X via Router 1.
Regarding the second question, LSAs 1,2,3,4,5.... can be advertised by neighbors, however this is based in specific circumstances. For example, LSA 2 is advertised by DRs specifying certain LSIDs, LSA 5 is advertised by Routers (ASBRs) redistributing routes from a Routing protocol to OSPF, LSA 3 are advertised by ABRs, and so on...
CISCO PRESS OSPF NETWORK DESIGN SOLUTIONS from Thomas M. II is a nice guide to reach a better OSPF understanding.
