905.2 Grade.
All vent and branch vent pipes shall
be so graded and connected as to drain
back to the drainage pipe by gravity.
(2006 International Plumbing Code).
This might prevent you from doing what you want, since your lower horizontal connection would have to slope toward the stack. Not sure if this is allowed or not, but it might be something to look into.
As DMoore suggested, running below slab is a viable option. Are you sure there are no lines running under the slab that may be closer?
The sink does need to be vented. Ideally, it should run up to an existing vent pipe that continues up through the roof. Some jurisdictions allow a mechanical vent which can be simply hidden in the wall, though the mechanical portion should be accessible for service and repair.
The drain slope is measured from the trap's outlet "invert", the lowest inside portion of the outlet pipe. Since pipe walls are parallel, you can actually measure the slope at any convenient surface, but for calculating if you have enough drop, use the invert. To tie into existing pipe, cut out a section and install a Wye or long turn Tee fitting. If none of the pipes can be moved to enable slipping into the hubs, the pipe can be cut farther up to enable gluing the fitting, then the butt ends are connected with a no-hub coupler.
As long as the pipes are waste pipes and not vents or something else, yes you can tie into them.
You should avoid pumps. But FYI, they are typically sort of a trash can with a pump inside. It could be placed on the floor in the storage room and the sink can easily drain into it. Such devices do get more complicated with floor level drains such as showers and toilets, but in your case, it's simple.
Best Answer
There will always be a vent stack in a home if you have plumbing in it. they usually poke out the roof to vent and go all the way to the basement and under the basement floor. All the waste water dumps into these pipes and exits the house to the main waste pipe outside your home. Depending on how many bathrooms you have the number of vent stacks can differ but most homes with one bathroom and a kitchen have two vent stacks. The bathroom vent stack will be the larger one (3-4 inches) and the kitchen is generally around a 2 inch pipe I believe. Usually when you tie into a vent stack you cut out a chunk and insert a "y" fitting of some sort. I am no plumber by any means but I have observed this being done. Also, when you tie into a vent stack there is a code in my state where you have to have the vent for the tie-in go above the fixture above it (i.e. above where the kitchen sink drain dumps into the stack if you tie into that stack) I think the name of that would be a branch vent ( see my simple diagram below). Good luck.