Extension of embedding from a compact submanifold into $\mathbb R^{n}$

Question

1. Suppose $M$ is a manifold, $N$ is a compact submanifold(with non-empty boundary) of same dimension, $f:N\rightarrow\mathbb R^{n}$ is an embedding into sufficiently-high-dimensional Euclidean space. Under what extra assumptions can we find an extension of $f$ to the whole $M$?

2. A possibly relative question: with $M$ and $N$ same as above, suppose in the following diagram $i$ is inclusion, $f$ and $g$ are embeddings. When is there an embedding $j$ that makes the diagram commute?

$\require{AMScd}$ \begin{CD} N @>\displaystyle i>> M\\ @V \displaystyle f V V\# @VV \displaystyle g V\\ \mathbb R^{n} @>>\displaystyle j> \mathbb R^{n} \end{CD}