Using mineral spirits on wood and then staining it?
1) As far as Mineral Spirits are concerned, you can apply stain as soon as the wood appears to be "dry" again (it will look "wet" until the mineral spirits have evaporated). However, see #3.

2) Most likely -- stain typically contains a thinner as a carrier, which dries more quickly in warmer temperatures. The age of the stain should not be a factor, unless it has lost the thinning agent due to the container not being sealed well -- this would make the stain appear darker. You can add back thinner to counter-act that. Thinner, in this case, is most likely mineral spirits -- you can use a retarder to slow the drying process if necessary.

3) The pre-stain more than likely has dried, curing the resins. A light sanding and a re-application within 15 minutes or so of staining should give you better results -- trying to re-stain after the drying and without sanding may cause the pre-stain to reject more of the stain than you'd like.