A few comments:
Lotion is thinner than cream. Creams tend to be too heavy to put on your face, other than at night.
Moisturiser describes the function, not the consistency : both creams and lotions can be moisturisers.
'Apply' is the more formal verb - the one that you'd find written on the label. Both 'on' and 'to' work. Informally, 'put' is the usual verb. If your skin is dry, you should use moisturiser/put on moisturiser.
1. My face is dry, I will smear cream on it.
No - 'smear' has negative connotations. In modern English, this word suggests an untidy or careless spreading of something possibly unpleasant - such as greasy or dirty marks. For example, you would complain if your hotel mirror had something 'smeared' on it. NB Think about the idiom 'smear campaign' i.e. spreading dirt.
2. My face is dry, I will rub cream on it.
Possible, but a bit violent. You might rub cream into dry elbows or feet.
3. My face is dry, I will put cream on it.
4. My face is dry, I will spread cream on it.
Not really. 'Spread' emphasises the idea of covering a surface. We spread butter on toast, or icing on a cake. We don't tend to use it for facial care.