Brief Answer:
Read below.

Detailed Answer:
I read your question carefully and I understand your concern.

In my opinion the first most likely possibility for your head tilting would be strabismus. People with strabismus often change their head position to adapt to their eye misalignement, in order to avoid seeing things double. Might have developed it since childhood.

The second possibility, a neurological condition, is cervical dystonia, or torticollis as it is commonly called. Dystonia can be primary, no cause can be identified, or secondary after a trauma or in the setting of generalized dystonia (not your case). What makes me put this possibility second and the strabismus first is the fact that dystonia usually is painful or at least feel some discomfort, while you say not to be aware of it.

Arthritis or other vertebral column issues also may cause it, but again would be painful.

As for whether it can be inheritable for your child, well that depends on the cause. If the cause is strabismus there is a genetic predisposition for strabismus, so it is a possibility.

I remain at your disposal for other questions.

---

Brief Answer:
Read below

Detailed Answer:
Hello again!

Yes if the ophthalmologist excludes that to be the cause then the next step is indeed to be seen by a neurologist for the possibility of dystonia which I mentioned.

As for strabismus inheritance, it is a complicated affair to offer an exact percentage because there are not only genetic factors but also environmental ones at play such as advanced mother's age, smoking during pregnancy, low weight at birth etc. It is known to have a genetic component from studies of siblings especially twins, but that only means there is a slightly higher percentage of risk for your child, not that it is going to happen. There is not a genetic test as it is not an issue of a single gene, there are many different genes contributing to it.

If it is due to strabismus, the cure depends on the degree of possible correction of strabismus. If something neurological like dystonia then it must first be studied and try to find a cause, a good medical and drug history is necessary, observation of the phenomenon, looking for other neurological signs, brain imaging may also be required. If it is confirmed to be dystonia and no cause can be found then there is treatment with injection of botulinum toxin, but I think you shouldn't run too far ahead. Can't be thinking about passing it to your son without determining exactly what it is either, so please try to not be so anxious about it.
It is not something to hurry and treat aggressively, it is not a medical emergency, you might have had it for a long time without noticing as you say. Movement disorders require careful examining and at times several follow up visits before making a diagnosis.

I hope to have been of help.

---