Generally, one has $\dim V//G=\dim V-\dim G$ but there are exceptions. For simple $G$ the exceptions have been classified in

Èlašvili, A. G. Canonical form and stationary subalgebras of points in general position for simple linear Lie groups. Funkcional. Anal. i Priložen. 6 (1972), no. 1, 51–62

Popov, A. M. Stationary subgroups in general position for certain actions of simple Lie groups. Funkcional. Anal. i Priložen. 10 (1976), no. 3, 88–90.

Generally, the quotient $V//G$ is singular. For simple $G$, the representations with $V//G$ being smooth (i.e. isomorphic to an affine space) were classified in

Schwarz, Gerald W. Representations of simple Lie groups with regular rings of invariants. Invent. Math. 49 (1978), no. 2, 167–191

Concerning your example $G=Sp(2n)$ and $V=L(\omega_1)⊕L(\omega_1)⊕L(\omega_2)$: Here $V//G$ is smooth of dimension $2n-1=\dim V-\dim G$ (see the paper of Schwarz above). The ring of invariants is freely generated by polynomials of multidegrees $(0,0,d)$, $d=2,\ldots, n$ and $(1,1,d)$, $d=0,\ldots,n-1$. The quotient map $V\to V//G$ is equidimensional (by another paper of Schwarz).

Generally, one has $\dim V//G=\dim V-\dim G$ but there are exceptions. For simple $G$ the exceptions have been classified in

Èlašvili, A. G. Canonical form and stationary subalgebras of points in general position for simple linear Lie groups (MSN English article). Funkcional. Anal. i Priložen. 6 (1972), no. 1, 51–62

Popov, A. M. Stationary subgroups in general position for certain actions of simple Lie groups (MSN English article). Funkcional. Anal. i Priložen. 10 (1976), no. 3, 88–90.

Generally, the quotient $V//G$ is singular. For simple $G$, the representations with $V//G$ being smooth (i.e. isomorphic to an affine space) were classified in

Schwarz, Gerald W. Representations of simple Lie groups with regular rings of invariants (MSN article). Invent. Math. 49 (1978), no. 2, 167–191

Concerning your example $G=Sp(2n)$ and $V=L(\omega_1)⊕L(\omega_1)⊕L(\omega_2)$: Here $V//G$ is smooth of dimension $2n-1=\dim V-\dim G$ (see the paper of Schwarz above). The ring of invariants is freely generated by polynomials of multidegrees $(0,0,d)$, $d=2,\ldots, n$ and $(1,1,d)$, $d=0,\ldots,n-1$. The quotient map $V\to V//G$ is equidimensional (by another paper of Schwarz).