GIT quotients for linear representations of $SL(2,\mathbb C)$

The answers are:

Question 1: yes.

Question 2: no.

Explicit linear generators follow from the first fundamental theorem (FFT) for $SL_2$. You can see my two answers to this MO question for an explicit proof of the FFT for $SL_k$. It is best to use a graphical language to represent these generators as in my article "On the volume conjecture for classical spin networks". J. Knot Theory Ramifications 21 (2012), no. 3, 1250022. This type of graphical representation is very old as you can see in this MO answer. Then the fun begins, namely trying to find polynomial rather than linear generators. Essentially this results from the Grassmann-Plücker relation. For $SL_2$ and forms of degree 1 or 2 this is easy to do by hand. For quadratics, the GP relation can be used to break cycles (the only thing produced by the FFT). In fact the article by Kempe in the second MO answer I mentioned does exactly that, with explanatory pictures. The first invariant which is not expressible by the ones you gave is for three quadratics corresponding to a 3-cycle containing each one of them. This is also the Wronskian of the three quadratics.

For quadratics an explicit system of generators which basically adds these Wronskians for each triple of forms is given in Section 256 "The quadratic types" in the book by Grace and Young.

The basic identity from that book needed for breaking cycles of length at least four is in classical symbolic notation: $$ 2(ab)(bc)(cd)(de)=(bc)(cd)(db)(ae)-(cd)^2(ab)(be)-(bc)^2(ad)(de)+(bd)^2(ac)(ae) $$ where $a=(a_1,a_2)$ etc. and $(ab)$ is the determinant of the matrix with first row $a$ and second row $b$ etc.

Using self-duality of $SL_2$ representations, the LHS can be a interpreted as a product of four $2\times 2$ matrices. This is basically the Amitsur-Levitzki Theorem for $2\times 2$ matrices.

I don't know how explicit you want to be but you can look at the article "Defining Relations for the Algebra of Invariants of 2×2 Matrices" by Drensky for more details. He treats the case of generic matrices while you are interested in matrices coming from symmetric bilinear forms by self-dualization. For generic matrices, traces of words of length 1, 2, 3 form a minimal system of $n+\frac{n(n-1)}{2}+\frac{n(n-1)(n-2)}{6}$ algebra generators. Drensky finds the polynomial relations between these generators. Here, with quadratic binary forms, the words of length 1 disappear.

For question 1, the answer is most easily described by thinking about $V^{\oplus n} $ as $ Hom(\mathbb C^2, \mathbb C^n) $. From this it follows that the projective GIT quotient $$ V^{\oplus n} //_{det} GL_2 $$ is isomorphic to the Grassmannian $\mathbb G(2,n)$. From this, it follows that the invariant ring $ \mathbb C[V^{\oplus n}]^{SL_2} $ is homogeneous coordinate ring of $\mathbb G(2,n) $. Your quadratic polynomials are the generators of this coordinate ring (and the relations are given by the Plucker relations).

Both questions are extensively dealt with in Weyl's book "Classical invariant theory" who investigated the invariant of classical groups on multiple copies of their defining representations. Determining a set of generators is called a "First Fundamental Theorem" (FFT) while the relations are given in a "Second Fundamental Theorem".

Question 1: This is best regarded as the action of $Sp(2n)$ on $V=\mathbb C^{2n}$ for $n=1$. Then, indeed, the ring of invariants is generated by all pairings $f_{ij}:=\omega(v_i,v_j)$. These are neatly organized in a $2n\times 2n$ skew-symmetric matrix. The relations are generated by all "principal" Pfaffian minors of size $(2n+2)\times (2n+2)$. For $n=2$ these are quadratic polynomials in 3 variables called the "Plücker relations". The quotient consists of all skew-symmetric matrices of rank $\le 2$ (which is, of course, the affine cone over a Grassmannian).

Question 2: In this case one is dealing with the group $SO(n)$ acting on $\mathbb C^n$ for $n=3$. Here things are a bit more complicated since $SO(n)$ is not strictly a classical group. The better problem is to look at the group $O(n)$ instead. In this case, the invariants are indeed generated by all pairings $p_{ij}=(v_i,v_j)$ which can be organized into a $n\times n$ symmetric matrix. The relations are generated by all $(n+1)\times(n+1)$-principal minors. In your case, the quotient would be the set of symmetric matrices of rank $\le3$. Since you are dealing with $SO(n)$ instead of $O(n)$ things are more complicated. In this case there are additional generating invariants namely all determinants of the form $\det(v_{i_1},\ldots,v_{i_n})$. For $n=3$ the quotient is the subset of $S^2\mathbb C^n\oplus\wedge^3\mathbb C^n$ given be two sets of relations: the rank conditions and the condition that the square of the determinant can be expressend as a Gram matrix $\det((v_{i_\mu},v_{i_\nu}))$.