partial correlation to control for a continuous variable when examining the relationship between continuous and categorical DV and IVs

I have two similar issues. I have two separate questions, one has a binary DV (is there a brain response yes/no) and the other a continuous DV (how widespread the response is). Each question is looking at the effect of a few IVs on each of the DVs. The IVs are age (continuous), head circumference (continuous) and sex (binary). I am working with infant data and as expected the age and HC are highly correlated.

What I want to do is look at the the effect of age and sex on the two DVs (separately) while controlling for HC. First, is the needed since I know HC and age are correlated (r=.74) so wouldn't controlling for HC of course remove any relationship that may have been there with age and the DV? Would it be best to look at the effect of HC and if there is none, leave it out as a covariate and instead interpret any age effects with HC in mind?

Also, which test is appropriate for this. Before controlling for HC I was using a logistic regression for the binary DV which allows for both continuous and binary IVs. For the continuous DV, I was using a linear regression. Is this even correct? I don't know which test allows me to input a continuous DV and both continuous and binary IVs, let alone a test that allows for that while also inputting a continuous covariate. I use SPSS but if anyone has a specific test in mind I should be able to find a way to do it if SPSS can't.

If you include both head circumference (HC) and age in your model you will be able to see if HC has an effect independent of age. This would be important if it was thought that having a big head made you more or less likely to respond even allowing for the fact that older infants have bigger heads. It would not be important if you thought HC was only important as another marker of age.

You can include any sort of variable as a predictor in your model (what you call an IV): binary, continuous, and categorical assuming your software will handle categorical appropriately. It does not matter what is on the left hand side of your model (what you call a DV) so it works for linear, logistic, and many others.

wouldn't controlling for HC remove any relationship that may have been there with age and the DV?

No. You can imagine multiple regression as trying to test the effect of age on the DV within same age groups: if age truly has an effect on the DV (that is not mediated through HC) it will not disappear after controlling. However, if HC and DV are highly correlated, it will be difficult to separate the age effect from the HC effect. See threads on multicollinearity, e.g. Is there an intuitive explanation why multicollinearity is a problem in linear regression?

Would it be best to look at the effect of HC and if there is none, leave it out as a covariate and instead interpret any age effects with HC in mind?

No. Univariate models can give different results from the full model (see any intro text on epidemiology, or e.g. Regression coefficients that flip sign after including other predictors). Such unreported testing also leads to model overfitting and inflated p-values (see any thread here on stepwise regression).

To comment more specifically on your situation: in early growth epidemiology it is customary to use age-adjusted measurements, so instead of using raw HC you could look up population-based growth curves and convert your measurements to Z-scores of HC for the corresponding age group. Just keep in mind that it slightly changes your model and interpretation.