Nuclei spontaneously decay according to a certain decay rate. There are however different kinds of decay, alpha, beta, gamma... What causes then the nuclei, when they decay, to do so in one way of another? Is there a different decay rate for each kind of decay?
It is conservation rules (mostly that of energy) that controls what decays are possible, and what channels are allowed (the strong interaction respected quark flavor, but the weak interaction does not). If the final state has higher energy than the initial state there can be no spontaneous process leading from one to the other.
The energy environment of a nucleus is a complicated place because it is affected by the strong nuclear force, electromagnetic forces and by the limits on state occupation imposed by Pauli exclusion. There are a lot of question already on the site about when various processes can proceed and related subjects
- $\alpha$ decay to more than one nuclear state
- What stabilizes neutrons against beta decay in a neutron star?
- How can a proton be converted to a neutron via positron emission and yet gain mass?
- Why is the (free) neutron lifetime so long?
- nuclear fission and half life
- Adding many more neutrons to a nucleus decreases stability?
Your question about the case where a nucleus has multiple modes have different half-lives is tricky. The measured result is the same half-life for each mode, but we can ask "If we could magically block modes and study the isolated half-life of each mode, what would they be?" If you perform the (often approximate) calculation for that question you generally find different half-lives. The interesting part is the the branching fractions for each decay are related to ratios of the calculated "isolated" half-lives, so you are seeing the effect of the different probabilities when you look at the frequency of different decays.
There is a different decay rate for each kind of decay. There is no specific "cause" for one decay versus the other. That is, given one nuclei, it is indeterminate what it will decay into after some given amount of time. However, we can determine the probabilities of what it will decay into. This is calculated using Quantum Field Theory.