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There are two primary reasons why BGP is deployed. First, it can handle very large route tables, e.g. the Internet's. Second, it generally provides much more "control" only what information is exchanged between routing domains and how BGP will chose a path for a destination. It's the latter reason why it's so often used between routing domains when there isn't a need for BGP scalability.
Is some ways BGP might be thought of as a simple routing protocol (super RIP) with the additional control of static and policy based routing.
Within a routing domain, an IGP is often being used even within large ISP. For them, their IGP of choice is often IS-IS.
There are several reason for using an IGP even when BGP is also used. Generally, IGPs converge faster. iBGP itself relies on interior reachability often provided by an IGP. You've asked about load balancing. BGP by default tends to prefer/use a single path where most IGPs easily support multiple path usage.
To your question about what a router does when it has more than one active routing protocol, there's Administrative Distance which provides the sequence of which routing protocol to trust first. By default, Cisco routers will use statics, and then eBGP, and then an IGP (with different rankings for those) and then iBGP.
