Hi MarcusCAG, We have had cockatiels for about 40 years and they all loved their head sritched with my nose. I think they like the warm air as you exhale on the back of their neck as you rub their head.
The male will be more vocal than the female and the female will chirp more than sing.
The lutino can be hard to tell male from female by looks but some say the dark ornge patch and a really yellow head can be a clue to a male.
A normal grey cockatiel male will have a very yellow head and dark ornge patch around their ears with a solid dark grey under side of tail feathers
The females head will be grey with very little yellow and a barbed marking under side of tail feathers.
And yes a female tiel can have eggs with or with out a male.
If you wanted little ones and you have a pair all you have to do is put in a nest box. I have had eggs spring, summer, fall, and winter.