The answer to your first question is “no”. The first counter-examples appear in dimension four - for example the complex projective plane. There are many simply connected manifolds and that are not aspherical that will fit the bill.

Here is a somewhat wilder example. Suppose that $V$ is a genus two handlebody of dimension three. For example, $V$ can be obtained by embedding a “eye-glasses” graph in three-space and taking a small regular neighbourhood. Let $M$ be the double of $V$ across its boundary. That is, we take two copies of $V$ and glue via the identity on the boundary.

The universal cover of $M$ is a copy of the three-sphere, minus a Cantor set.

Finally, I don’t see any path to an answer to your second question.

The answer to your first question is “no”. The first counter-examples appear in dimension four - for example the complex projective plane. There are many simply connected manifolds and that are not aspherical that will fit the bill.

Here is a somewhat wilder example. Suppose that $V$ is a genus two handlebody of dimension three. For example, $V$ can be obtained by embedding a “eye-glasses” graph in three-space and taking a small regular neighbourhood. Let $M$ be the double of $V$ across its boundary. That is, we take two copies of $V$ and glue via the identity on the boundary.

The universal cover of $M$ is a copy of the three-sphere, minus a Cantor set.

I don’t see a path to an answer to your second question. Perhaps you would be interested in Thurston’s geometrisation programme. It is sometimes described as a version of uniformisation in dimension three. It sadly does not generalise to dimension four.

Here is a first counter-example.

Suppose that $V$ is a genus two handlebody of dimension three. For example, $V$ can be obtained by embedding a “eye-glasses” graph in three-space and taking a small regular neighbourhood. Let $M$ be the double of $V$ across its boundary. That is, we take two copies of $V$ and glue via the identity on the boundary.

The universal cover of $M$ is a copy of the three-sphere, minus a Cantor set.

The answer to your first question is “no”. The first counter-examples appear in dimension four - for example the complex projective plane. There are many simply connected manifolds and that are not aspherical that will fit the bill.

Here is a somewhat wilder example. Suppose that $V$ is a genus two handlebody of dimension three. For example, $V$ can be obtained by embedding a “eye-glasses” graph in three-space and taking a small regular neighbourhood. Let $M$ be the double of $V$ across its boundary. That is, we take two copies of $V$ and glue via the identity on the boundary.

The universal cover of $M$ is a copy of the three-sphere, minus a Cantor set.

Finally, I don’t see any path to an answer to your second question.