Let's say I am running Cox proportional regression model for a variable with 4 levels (strongly agree, agree, disagree, strongly disagree). When I code it as an ordinal variable with these 4 categories, it is an insignificant contributor. However, if I make this variable binary (agree vs disagree), it is significant. Is this still appropriate? When is it okay to convert an ordinal to a binary variable? How do I go about interpreting this result? Thank you!
Even if you tell software that a predictor is ordinal rather than numeric/scale (which you used isn't completely clear from your question or comment), its handling would generally still be based an an assumption of equal spacing between levels. See, for example, the UCLA web page on how R handles ordinal predictors. A failure of that assumption could lead to the result you found.
As @Alex suggested in comments, the best approach with so few levels might be to treat it as a nominal predictor to get around that assumption, and evaluate the significance of the predictor as a whole rather than relying on tests of individual associated coefficients. For example, do a likelihood-ratio test of two Cox models, one with the predictor and one without.
This could be a case of the difference between significance and non-significance itself being non-significant. See this paper: http://www.stat.columbia.edu/~gelman/research/published/signif4.pdf