If you think the self comes from memories and that cognitive biases affect memories, then what does that say about the objectivity of the self? Do you think we are rational and objective; emotional and subjective; or a combination of both?

The self might be the product of cognitive biases and subjective, erroneous memories. Nonetheless, we are more than emotional and subjective. We also have the capacity for a limited degree of rationality and objectivity.
Research by Daniel Kahneman and other psychologists has shown that much of our thinking is automatic and intuitive. We don't consciously deliberate, analyze, and calculate answers. Instead, unconscious processes do the work for us by noticing trends and making use of quick-and-dirty "rules of thumb." Feelings and associations simply pop into our minds.
This helps explain why people are prone to making errors based on cognitive biases. The intuitive system (which Kahnmen calls "System 1") makes us jump to conclusions. Sometimes it even makes us blind to evidence that would refute our perceptions and beliefs.
This also suggests that our personal memories -- what we believe has happened to us and what we believe about our past behavior -- are based on faulty, biased, and subjective information. 
Work by Elizabeth Loftus and her colleagues has confirmed that our episodic memories are sometimes highly unreliable. Moreover, in recent years, research has indicated that we "overwrite" our old, episodic memories every time we replay them. The original memory gets replaced by the story we retell, allowing for errors and embellishments to creep in. In compelling experiments, Loftus has even shown that we easily can "implant" false childhood memories in others, simply by telling them that the event occurred. Presumably, many of us acquire such false memories as we go through life by inadvertently internalizing stories we've heard or associating other people's experiences with our own.
If you consider the combined shortcomings of our episodic memories and System 1, it's hard to argue that the self is either very objective or highly rational.
We're also capable of turning our conscious spotlight on problems and trying to reason about them in very deliberate, rational ways. Kahneman calls this slower, more self-conscious set of processes "System 2," and these are the processes that some people use to craft logical arguments, question intuitions, test assumptions, and weigh evidence. People can learn about logical fallacies, cognitive biases, and the unreliable nature of our episodic memories. They can't eliminate the effects these have on the self, but they can remind themselves to check their feelings and consider other points of view. This permits us to correct for some of our subjectivity and think rationally at least some of the time.
https://staff.washington.edu/eloftus/Articles/sciam.htm