I have a familiar and I want to give it something and then dismiss it to its pocket dimension while it's hold the item. Obviously, I want the item to go into the pocket dimension with the familiar and come back out again with the familiar. Any rule saying I can't do it?

Talk to your DM. 

Only the familiar goes into the pocket dimension.

The reason is simple, being to use it for safe storage like that is basically giving you the effect of multiple higher level spells. As an already powerful 1st level spell, allowing that would be way too much.

But yeah, you can talk to the DM.

I think it depends more on what the item, specifically, is that you're trying to tuck away.  If it's just a Tiny mundane item, who cares?  If it's something like a weapon, well...  that may be reasonable, depending on the weapon in question.  It gets messier from there.

And all of this is at the discretion of the DM, so you should really be talking to your DM about this instead of us.

My familiar has a pair of eyeglass rims (no glass) fused to his head.  I use as an object to cast Light on that the snake can use as a flashlight.  When I change the shape of the familiar the rims show up attached to odd places.  The octopus looks like a narwhal etc.  

But no I would not ask the DM to allow summoning him inside the locked jewelry case and then having him bamf out with the booty. 

Like others have said, it is up to your DM. 

I would say no. It is a spirit, and when it is reduced to zero hit points it disappears, leaving no physical form. I would assume it is similar when you dismiss it to the pocket dimension. It disappears with no physical form so cannot carry anything.