Is lateral thinking a valued trait in engineering, or to put it another way can you get through an engineering degree and whole career by memorizing everything you have been taught but have no ability to think outside the box?
Yes, and yes.
Define the problem a bit more clearly:
Does an increased application of "lateral thinking" (definition pending?) result in higher wages/salaries in engineering fields?
I would think so. We can measure this quantitatively, at least in terms of outcome, and, given access to suitable data; filtering those data by the antecedent may be less straightforward, however.
But you also asked another question:
Can you complete an engineering degree, and whole career, using wrote information?
I also think so. I have met plenty of engineers, who are by all practical and technical definitions "engineers", and who aren't especially keen on outside-the-box thinking; everything they do is either as prescribed by textbooks, appnotes, etc., or passed down as given from on high.
Regarding math, I think so as well, but not to the vaunted level we would like to think -- 99% of engineering consists of busywork/housekeeping (project management, office politics as applicable, part and supplier selection, etc.), basic arithmetic (evaluating equations as given in appnotes / etc.), and informal combinatorics (finding the right set / permutation / sequence to more-or-less satisfy some largely-incomplete constraints/specifications). Even for the most erudite among the most common types of engineer, calculus and such makes up only a very small fraction of overall activity. Less common types can use more -- for example, sooner or later, someone needs to write the simulation tools used by others, which will involve field equations, differential equations, and vector calculus, so, implicitly, calculus and linear algebra as well. Though it's probably arguable that again, 95%+ is spent on combinatorics: writing code to implement those functions.
I can probably reduce that further and just say it's all combinatorics, as that's ultimately what squishy neural networks are good at; that, and detecting (pattern matching) environmental cues to, you know, learn all of language and society and how to actually do a job, as presented by other squishy-neural-networks.
Tim