The configuration you pictured is not really a good idea. Waste water from the washer is going to flood the heater drain pipe, leaving nasty residues that could conceivably siphon back into the heater if connected to the heater drain valve. Not likely, but possible any time two drains are connected at different levels other than directly into the vented out flow pipe. The other problem will be that if the drain is a open pan drain only, washer waste water will back up into the pan. Washers tend to eject water quickly and can partially fill the pipe above the trap.

I would rather see you pump it up to the inlet of the washer drain.

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.