Does it change anything in a multilevel model when the predictor is a higher-level variable?

I am working on a project where I have to study the link between hospital competition and mortality (but also the duration of hospital stays). More precisely I want to determine whether the more a hospital faces competition the better it performs, and therefore the lower the mortality rate. There is an index (Herfindahl-Hirschman Index or HHI) that can be used to calculate the degree of competition a hospital faces. It can be calculated for each hospital. I am looking for an appropriate model to study this link. My first thought was a multi-level logistic regression model. So I want to proceed according to the following equation (That's just illustrative, it's not a mathematically correct equation:

Mortality= Intercept + a*Patient characteristics + b*HHI+ c*Hospital's other characteristics+ residuals.

Mortality (coded by yes or no) and patient characteristics (age, sex, diagnosis, severity of disease, etc.) are lower level variables. The Herfindahl-Hirschman Index (the hospital-related predictor) and other hospital characteristics (hospital status, overall volume of cases treated by the hospital) are higher level variables. In a multilevel model, predictors are most often lower-level variables (individual level), but in my case, the predictor is a higher-level variable. Does this change anything in the multilevel model? Or is another type of model more appropriate for my analysis? Finally, if my outcome variable is a continuous variable (such as length of stay, in number of days), do I need to run a multilevel linear regression?