For FindRoot the precision of the initial guess does not matter. The initial guess need not satisfy anything, since it will be updated. It is the equation that must be satisfied by a solution, and it is the equation that should have a sufficient precision (that is, at least WorkingPrecision). Whatever the precision of the initial guess, there is no problem in changing its precision to the WorkingPrecision. The adjustment in precision could be considered a preliminary step or update to the initial guess.
For NDSolve, it is the whole system that must be satisfied: the ODEs and the BCs/ICs. Thus the whole system should have a sufficient precision, at least that of WorkingPrecision.
Finally, both commands issue a warning when the appropriate equations have insufficient precision, but they promote the precision of the equations to WorkingPrecision nonetheless. The warning just points out that the user is asking for something that does not make sense on the face of it: The insufficient precision might be due to a user mistake, or, in the case of a computed equation, to too great a loss of precision in computing the equation.