Is there a "beta distribution" over the entire real line? The malleability of the beta distribution pleases me; it can be symmetric, asymmetric, platykurtic and so on, as the following picture shows us:

I thought it would be interesting to use it to model the errors in regression problems, but its support is only defined to be $(0,1)$ so I would like to know if there is a "beta" distribution with support $(-\infty,\infty)$ because a error lying on $(0,1)$ is such a strong assumption.

It seems to me the convex shape the beta distribution can take is a result of the bounded sample space.  If we move to a sample space that is bounded on one side then we are dealing with distributions like gamma, Weibull, etc.  If we move to an unbounded sample space then we of course have the normal distribution.
All of these distributions, including the beta and scaled beta, can be used in regression via generalized linear models.  Rather than developed around an "error term" these models are typically developed by expressing a parameter in the likelihood or probability distribution in terms of a linear predictor.  You could re-parameterize a model in terms of its mean or median and express this in terms of a linear predictor.
