Since you titled your post to be about venting, here is a diagram with the laundry tub and pump basin vents done according to my local code requirements (IPC / IRC). Your code requirements may be different. Also important is to get a check valve installed in the pump discharge line. That s-trap is most likely not serving any pupose at all. The water in the verticle pump discharge will flow back down and fill up the basin some every time the pump stops running, most likely siphoning all the water out of the s-trap at the same time. Although the pump and your drians probably function fine as they are, you should get them vented to the outside air for health reasons if none other. That basin will not completely empty out and will hold some water, dirt, scum for mold and bacteria to grow in, and then be released through the open vent in your basement.
As for the leaks under the kitchen sink.... are they leaking while the pump is running and the sink is not being used? or do they only show a leak when the sink faucet is running? Perhaps there is a partial blockage in the main drain line allowing the pump discharge to back up and leaking through faulty connections under the sink. No matter, the leaking connections under the sink need to be repaired or replaced as they should not leak for any reason.