I think y+ is crucial for accurately modelling boundary layer by means of wall-functions under RANS frame. For engineering flows, grid-irrelevance check is a must nowadays. I think grid-irrelevance is case dependent and there is no accurate means to predict a proper grid size so far. For LES, its boundary layer treatment is different from RANS, commonly y+ is much smaller than RANS.

For RANS, near wall grid spacing is depended on the near wall treatment in your choice. A range of y+ 1.0 to 5000 is likely not right when using the log law in the standard k-eplison model. When use the log law, you should have your y+ larger than the buffer layer, say y+>30. When use a two-layer log law, you could have your y+ near 1.

You should be very cautious when you're modelling surface roughness. Some studies claims your near wall y+ should at at an order of 0.001 !

When using k omega model, the omega wall boundary condition is a singularity problem that will cause numerical accuracy with a poor near wall grid spacing. Concerning only the y+ regardless of the normal grid expansion ratio is also wrong.

There are more than near wall grid spacing you should concern. The near wake after a body, separation area, laminar to turbulent transition area should be refined to check whether if the steep change in flow variables has been well presented by your discrete mesh elements.

You can't save or say "minimize" the work on grid independent study for a new problem. Only for a similar problem you have dealt before, you are likely to apply your previous experience to obtain the proper grid in a shorter time.

This book is suggested if you would like to go further: Patrick J. Raoche, Fundamentals of verification and validation, Hermosa publication, Socorro, New Mexico, USA, 2009, ISBN 978-0-913478-12-7.

If other equation than turbulence one is to be solved, you must consider of course the associated Peclet number which compare convection transfer Vs Diffusion transfer. This will allow you to estime the transfer phenomenon boundary layer thickness. The rule to obey is "you need more than one mesh inside the real boundary layer". So your mesh dimension must be chosen according with this rule.

I would like to add to above discussion that usually 8-12 mesh inside boundary layer actually is enough. the Idea of Peclet number is interesting, In fact if you place a particle in your finest cell it should not reach the other side of that cell in a single time step by convection/diffusion . this will help to estimate the early grid size, but grid study , whenever its applicable, is inevitable!

No. Just repeat your runs for finer and finer grids until your solution does not change anymore anywhere in the field.( Needless to say that your first grid must account for all physical aspects of your problem).

Yes, as said earlier, i think, mesh adaption is the optimum tool for getting an optimized mesh size. Moreover, it could also help in reducing the required computation power.