Is it always true that the adjusted OR will be lower than the crude OR?

No. The effect of adjusting depends on which direction the confounder influences the association. If the confounder influences the exposure (E) and the outcome (O) in the same direction, this would increase the strength of the E-O association, and adjusting would attenuate it (remove the confounding). If the confounder influences E and O in opposite directions (e.g. increases the chance of the outcome while lowering the chance of being exposed), the confounder would attenuate the observed E-O association and adjusting would increase your effect size.

How does one do the actual MV analysis? With a list of univariate OR’s? Or can it only be done with the original dataset?

You have to put in the original data, the univariate OR’s are not used for the adjusted analysis.

In R you could add covariates with +. For linear regression, the code would look something like lm(x~y + a + b, data = data), where x is the outcome, y is the predictor and a and b are covariates.

Which function in R does this?

Depending on what type of analysis you want to do, there are many options. For logistic regression, you could check out the glm() function. For linear regression lm() does the trick.