Awesome question! Thanks for asking it and giving me the motivation to look back at the literature to increase my own understanding!
I do want to point out one issue of semantics before starting. Just to avoid confusion, I'm going to follow the example of some of the well-known experts in this field and distinguish dendritic spikes from action potentials. The main difference is that a voltage spike in a dendrite or dendritic spine will experience attenuation (essentially a weakening of the spike) as it propagates toward the cell body, and it is distinct from the all-or-none voltage response of an action potential. It is generally accepted that action potentials are initiated in the axon (hillock) because that is where dendritic and somatic spikes will ultimately sum to fire the all-or-none voltage response. I just want to clarify what I mean by dendritic spike when I use it during this explanation.
What happens at the branch? So, to answer your first question, you're absolutely correct that when a dendritic spike occurs, that spike will encounter multiple branch points along the dendrite as it proceeds toward the soma. When a spike propagates to a branch point, the properties of the branches determine what happen to the spike. One important property is referred to as the geometric ratio (GR). This property was described Goldstein and Rall in 1974 (manuscript here for review https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1334570/) and the gist of the equation is that spike attenuation is dependent upon the diameter of the branch the spike is coming from and the branch the spike is going into. The concept behind this is called impedance mismatch, but that's complex and I won't get into it. Just know that spike propagation is favorable from a large to a small branch, but is not favorable from a small branch to a large branch. If you wish to read more about this concept, I'd recommend the Goldstein and Rall paper referenced earlier and chapter 14 of the textbook "Dendrites" by Spruston, Stuart, and Hausser.
Additionally, the presence of leak or voltage-gated channels in that branch can determine how well the spike will propagate or how much attenuation will occur. This is an easier concept to grasp because the idea is essentially that more channels open when the spike passes through that branch, the more charge leaks across the membrane and causes attenuation of the spike.
All questions regarding preference of flow To sort of attempt to answer all your other questions at once, nothing really forces the spike toward the soma or a specific axon terminal. There are dynamic changes in membrane potential that may prevent spike propagation down one branch or action potential propagation in a certain direction (like how action potentials aren't thought of as propagating from the axon to the soma because of the hyperpolarization and long inactivation of sodium channels at the area that just fired, preventing what is called backpropagation at that particular time).
I think a little reading on neuronal cable theory might be of some help with this question too. This link is to a PDF that gives a pretty brief explanation of this concept https://nanohub.org/resources/20112/download/Lecture1_Passive_Conduction.pdf. This is a pretty math-heavy field, so beware if math isn't your favorite subject!