It's difficult to say exactly what's going on without seeing your model and data. One explanation is that your model is not appropriate for the data. Alternatively, your data may not be sufficient for identifying all of the parameters in your model.
To address this issue, you could try to develop a different (simpler) model, try to estimate fewer parameters in your model, or (hardest of all) try to identify what additional data you need to collect for fitting this model and then collect this data.
I would start by trying to figure out which parameter(s) lead to the first error and focus on those. One thing that might help with that is to do a pooled fit (nonlinear regression rather than mixed-effects). That's essentially what happens during the RefineBeta0 step. If you find that one parameter has a very large standard error or has a strange value, that's likely the one that is causing problems.
The second error is harder to address. Sometimes this happens because parameters take on extreme values during fitting. The most straightfoward way to prevent that is by putting bounds on the parameters you want to estimate, but our mixed-effects estimation doesn't curretnly support bounds. But you could potentially work around that by adding something to your model that functions like parameter bounds. For example, you could introuce a new parameter and assoicated initial assignment rule that enforces bounds: k_out = min(max(k_in,k_min),k_max);
