Does our universe have to be embedded in a higher dimensional space?

The universe being open, closed, or flat only determines the type of geometry one must use to describe distances (and time). For open and closed geometries, Euclidean Geometry is not what one should use. I would also agree that our universe being open, closed, or flat has nothing to do with the number of dimensions it contains.

At present, there is no direct evidence that there are any additional dimensions over 3+1 dimensions (this just means three spatial dimensions and one time dimension). However, many GUT theories do include additional spatial dimensions in an attempt to unify all of the forces.

Also at present, we have very good evidence that the universe is 'flat'. What this means is that the angles of a triangle have to add up to $180^{circ}$ and distances are measured in the standard Euclidean way. When we talk about the universe being flat, this is a purely global statement. Locally, however, it is completely possible to live in curved space. We actually do live in curved space. The mass of the Earth is curving space and time in a way that General Relativity predicts, and therefore clocks run very slightly differently depending on where you are on the Earth's surface, and distances are very closely approximated by Euclidean distances, though they're not.

How do we actually determine that the universe is globally flat, you may ask? We use what's known as a standard ruler. Much like a standard candle, if we think we understand the physics, any deviation from what we would predict gives us new information about the universe (in the case of supernovae, it's that they can be used to measure distances, and therefore map out the expansion of the universe). We use the angular size of fluctuations in the Cosmic Microwave Background Radiation (we think we understand the physics behind the fluctuations fairly well) to test how much, if at all, the universe deviates from flatness. The latest results from Planck show very good agreement with the standard picture provided to us by the standard LCDM cosmological model.

Below is the power spectrum of the temperature fluctuations of the CMB. The location of the first peak is what cosmologists use to measure flatness.