Does smooth section of a quotient space $G/H$ define an immersion?

Question

Question 1:

Let $G$ be a Lie group and $H<G$ a Lie subgroup of $G$, given the projection $p:G\to G/H$ and a smooth (local) section $s:G/H\to G$ s.t. $p\circ s=id$, then does this imply that $s$ is an immersion? So a section can in general be considered a submanifold of $G$? When will $s$ be an embedding?

Question 2:

If the first case is true, denote the image of $s$ by $M$. Then is it true that the pointwise multiplication $M\cdot H=G$ (at least locally)? On the other hand, if $M$ is given that satisfies $M\cdot H=G$, is it true that $M$ is always the image of a smooth section? Is there a standard way to construct and classify submanifolds of $G$ that satisfies $M\cdot H=G$?

Answer 1

Question 1: $p:G\to G/H$ is a principal fiber bundle with structure group $H$. Thus $s$ is an immersion (since $p\circ s =Id$, $s$ has rank $=\dim(G/H)$, and $s$ is injective and a closed embedding).

Question 2: Yes, $M.H=G$. But not every $M$ with this property is the image of a section. $M$ might be a leaf of the horizontal foliation given by a flat principal connection, then $p:M\to G/H$ would be a covering mapping.