What do we hope to learn from the LHC?
A few things. The first and foremost is "What broke electroweak symmetry?" That is, what is the explicit mechanism where by our world settled into a state where the electroweak force appears as two separate forces. The most commonly used mechanism in our theories is the Higgs field, which basically provides a background against which the electromagnet force and the weak force appear separate. This is similar to how the presence of an atmosphere on earth can make momentum appear not to be conserved.
However there are several other supposed mechanisms and the LHC will test them as well. Unfortunately the LHC has been misreported as a machine built to find the Higgs Boson, but that is not its aim. Finding the Higgs boson would mean that the simplest mechanism of electroweak symmetry breaking is the right one. However not finding it wouldn't be some kind of disaster.
After that the next major thing would be detecting the presence of supersymmetry particles. Supersymmetry is basically the biggest symmetry a quantum field theory can have and its existence would imply new particles.
After that it may be capable of telling us if there is extra dimensions or put limits on how large extra dimensions can be.
What predictions borne from current scientific models might be verified or refuted?
Several versions of supersymmetry, certain mechanisms of electroweak symmetry breaking and theories with large extra dimensions.
Will the LHC experiments tell us anything new with regard to the very early universe, string theory, branes and the actual origin of the universe?
The discovery of supersymmetry particles would indicate string theory is on the right track. As for the origin of the universe, the LHC may tell us what material makes up Dark Matter, which is important for cosmology and theories of the early universe.
Can the findings of the LHC do anything that will provide further positive evidence for the BBT and current cosmological models?
That wouldn't be its direct or expected contribution. Rather it will improve our understanding of particle theory, which may then be used to aid the study of cosmology.