Rosewood and yes, lemon oil is fine but one of the specialty guitar products is a bit more effective. As long as what you're using doesn't have silicon in it, you're good.
We talking about the fretboard? Looks like rosewood to me. Yeah lemon oil should be fine
that looks really dirty. if i were you i would pre clean with rubbing alcohol on a rag thats more dry than wet with the alcohol. get it till it looks clean enough and then do a final clean with a very small amount of lemon oil. remember more dry than wet for both substances when applying them to the rag.
I like using the Meguiars kit that fender sales. I have a bottle of lemon oil too but I use that stuff to treat the wood more than to clean it. Thats just a suggestion, but you can clean it with that too.