A little late to this but in case you didn’t get the answer yet:

Testosterone is a lipid hormone as you said. When it’s in circulation it’s usually bound to a transport protein like albumin or sex hormone binding globulin. Some testosterone is free and not bound to a transport protein. Lipid hormone receptors are generally intracellular/cytoplasmic but I think there is some evidence they can be on the cell membrane. Generally they’re taught to be intracellular though.

When testosterone binds the androgen receptor, the steroid-receptor complex transports to the nucleus and binds androgen response elements on the DNA (I.e. it acts as a transcription factor) and leads to transcription of certain genes. There isn’t a classic second messenger in this case.

Second messengers are usually downstream of G protein coupled receptors. For example, follicle stimulating hormone will bind the FSH receptor which is a transmembrane GPCreceptor that leads to activation of cAMP, a second messenger that activates PKA to alter gene expression and protein function.