Q1: Do have all closed, oriented manifolds a non-zero vector fields with infinitely many isolated singularities?
Q2: Do have all closed, oriented manifolds a non-zero vector fields with finitely many isolated singularities?
Q3: Do have all closed, oriented manifolds a non-zero vector fields with non-isolated singularities?
In regards to Q3, I think it is possible to construct a vector filed using bump functions which is zero in a open neighborhood. Am I right?
One does not say a vector field has a "singularity" (unless it fails to be defined at some point, maybe). One says that a vector field has a "zero", a point where $V_x = 0$.
On to your questions.
No, you have assumed the existence of an infinite discrete set, which contradicts compactness.
Yes, this follows eg from a variant of the transversality theorem. The argument is as follows: Given a vector field $X: M \to TM$ one may perturb this map just so slightly to be $Y: M \to TM$ so that $Y$ is transverse to the zero section. However, $Y$ may not be a section; but $f = \pi Y$ is a map $M \to M$ which is close to the identity, hence a diffeomorphism. Then $(df^{-1})_* Y$ is the desired vector field on $M$ with isolated singularities. Because $M$ is compact there are finitely many.
Sure, as you say pick some nonzero vector field, pick $x$ and $y$ points where this vector field is nonzero, and take $\rho X$ where $\rho$ is a bump function which is nonzero at $x$ but zero at $y$.