Couple things come to mind here. In no particular order.
The rootsing thing, I have had a few Brahmas that have shown no interest in roosting. Just figured that's how they were wired.
The age of lay varies considerably in various lines. A general rule of thumb, the bigger the bird, the later onset of lay.
Another factor here is time of year. The fact that days are getting shorter is likely affecting start of lay. Good chance this same age pullet would lay sooner if it was spring and the days were getting longer.
Your concern about her weight is well founded. A very fat hen will often quit laying all together. Might be a factor here as well.