We have very good evidence that the observable universe had a beginning. Gravity, and the other fundamental forces, acting on a slightly clumpy gas seems to be enough to produce stars, galaxies, and planets, based on computer simulations.
Mutations are observable, natural, and all but inevitable. Whether they are guided in a specified direction beyond “What survives?” is a philosophical question, not a scientific one.
We see plenty of transitional forms. Claiming that they cannot exist, and then drawing arbitrary lines on phylograms, or making horribly inaccurate claims about relationships, to separate them doesn’t make them go away. A few examples include: myllokunmingiids; stem-tetrapods that have lateral lines and labyrinthodont dentition, like certain fish, but no modern amphibians; Pakicetus, which has clearly amphibious features, and Basilosaurus-like dentition; and the numerous mollusks in the Waccamaw Formation (as an example familiar to me), whose appearance is halfway between their Yorktown-Formation predecessors, and their recent equivalents. Really, nearly every fossil in any deposit is transitional, we just have so few really good deposits that we don’t have a complete transition in most cases.
Scientifically, it is impossible to prove or disprove the existence of anything non-physical. One can produce scientific evidence that can say “this claim about the physical actions of this spiritual entity is possible, given our knowledge about the physical world” or “this claim about the physical actions of this spiritual entity is does not accord with our knowledge of the physical world”, but those are not proof-statements.