Absolutely. Possessive words are very often left out in Latin when they are not necessary. So whereas in English I have to say "my head hurts", in Latin, as in French, you essentially say "the head hurts", since it can be assumed that it is my own head that I am talking about.
Yes. Je me lave les mains = I wash (myself) the hand, when the English says "I wash my hands".
But the possessive was already implied in the reflexive (me: myself)
In "J'ai mal à la tête", (litt: I have a pain to my head) as I am the person who speaks, It would be really hard to suffer a pain in someone else's head...
So we don't feel the necessity to add more details. French language doesn't like pleonasm and tautology, but it doesn't skip the possessive as easily than Latin.
The most common sentence structure in Latin is Subject-Object-Verb (which you will often find referred as "SOV" in the forum).
Since word order is pretty fluid in Latin, you can find other structures, which usually serve to place some kind of emphasis on the words that are being "misplaced".
And there a few exceptions, for example with copulative verbs...