Maybe Andre has answered your question. I can say that your questions seem a bit too open-ended for precise answers.
Meanwhile, I can recommend $p$-adic Numbers by Fernando Q. Gouvea. I have the second edition.
With that in mind: first, by Ernst Selmer, 1957, we find that the cubic (homogeneous) form $ 3 x^3 + 4 y^3 + 5 z^3 $ has nontrivial zeroes in every $\mathbb Q_p$ including the reals as $\mathbb Q_\infty,$ but has no rational root except $(0,0,0).$ That's actually not in this book. By root I mean $ 3 x^3 + 4 y^3 + 5 z^3 = 0.$
Second, given a quadratic form (homogeneous degree 2) that has non-trivial roots in every $\mathbb Q_p,$ then it has a non-trivial rational root. This is the celebrated Hasse-Minkowski Theorem, Gouvea page 79, Theorem 3.5.2.
Third, given a form $f$ and a representation question $f(x_1, \cdots, x_r) = n,$ what you do is create $ g(x_1, \cdots, x_r, x_{r+1}) = f(x_1, \cdots, x_r) - n x_{r+1}^2. $ Then you ask whether $g$ has nontrivial roots. This leads quickly into unfamiliar territory and the word isotropic. See J. W. S. Cassels, Rational Quadratic Forms, up to page 63. Cassels lists the 2-adic squareclasses carefully on page 43, in the context of the Hilbert norm residue symbol.
Finally, page 73 in Gouvea, Problem 116: Show that if $b \in \mathbb Z_2,$ and $b \equiv 1 \pmod {8 \mathbb Z_2}$ (so that in particular $b$ is a 2-adic unit), then $b$ is a square in $ \mathbb Z_2.$ Conversely, show that any 2-adic unit which is a square is congruent to $1$ modulo $8.$ Conclude that the group $\mathbb Q_2^\times/ \left(Q_2^\times \right)^2$ has order $8,$ and is generated by the classes of $-1,5,$ and $2,$ so that a complete set of coset representatives is $\{ 1,-1,5,-5,2,-2,10,-10 \}.$
Well, that's a start.