do we really need a human-level AI to start the Singularity, or will an AI with animal-level intelligence suffice?
The requirement from "theory" of the singularity is that:
If both these things hold, then each generation of AIs will continue to improve. This is often assumed by singularity pundits to be an exponential growth curve e.g. each iteration makes +50% compound improvement on whatever measure is being made of intelligence. (Aside: Personally I find this a major weakness in the argument for the singularity being meaningfully possible, that it assumes this growth)
The first item of these two is important to your question. For the singularity to work, there is a baseline capability required - the AI needs to be able to design and build other AIs. A general intelligence at the level of an animal - at least any animal intelligence that we are aware of - does not seem capable of this task. It is not clear that humans are even capable of this task when the AI being built has to possess at least some general intelligence.
The term "animal-level intelligence" is tricky. The narrow AIs that we currently build can outperform animals and humans on specific tasks, but in terms of general intelligence they do not score highly (or at all). If we could build one that can outperform humans on a "building an AI" task, it might still have the general intelligence of an animal whilst having the capability to bootstrap an iterative process of self-improvement. This does seem like a very dangerous experiment to try though, with idiot-savant-deity AI and paperclip maximiser scenarios as possible outcomes because the AI's general intelligence lags behind its raw capabilities.