I have to say, for something like this I can't think of a single situation where there'd be a benefit in wet-sanding the primer with 1000 grit!
Alice, when you paint you can get away with much coarser sanding scratches than when using a clear finish where you see through to the wood's surface. Even the sanding for varnish only needs to go up to somewhere around 180 or 220 in most cases, and when painting you can get away with final sanding to 120 or 150 without any problem.
And no, you don't need to sand between coats of primer and paint, or between coats of paint. You can sand if you need to remove any drips or runs, if there's a slight roughness for whatever reason (sometimes the grain stands up after the primer is applied, if dust has settled in the drying paint) but other than those things as long as you don't wait to long between coats you shouldn't need to sand again after finishing starts on something like this.
Alice, when you paint you can get away with much coarser sanding scratches than when using a clear finish where you see through to the wood's surface. Even the sanding for varnish only needs to go up to somewhere around 180 or 220 in most cases, and when painting you can get away with final sanding to 120 or 150 without any problem.
And no, you don't need to sand between coats of primer and paint, or between coats of paint. You can sand if you need to remove any drips or runs, if there's a slight roughness for whatever reason (sometimes the grain stands up after the primer is applied, if dust has settled in the drying paint) but other than those things as long as you don't wait to long between coats you shouldn't need to sand again after finishing starts on something like this.