Over time the surface of the plastic part that joins the tank to the bowl can get tiny defects in it that prevent the flapper from making a good seal. As Jeff suggests, you could try cleaning that part, or just replace it.

Yes, you can plumb the outlet of the condensate pump into the tailpiece of the shower. Assuming the condensate line is properly trapped, then there should be no problem.

You'll notice in the code snippet below, that it says "If discharged into the drainage system, equipment shall drain by means of an indirect waste pipe.". Which means you cannot, plumb the condensate drain directly into an untrapped waste line or vent.

Uniform Plumbing Code

Chapter 8 Indirect Wastes

814.0 Condensate Wastes and Control.

814.1 Condensate Disposal. Condensate from air washers, air-cooling coils, fuel-burning condensing appliances, the overflow from evaporative coolers, and similar water-supplied equipment or similar air-conditioning equipment shall be collected and discharged to an approved plumbing fixture or disposal area. If discharged into the drainage system, equipment shall drain by means of an indirect waste pipe. The waste pipe shall have a slope of not less than one-eighth (1/8) inch per foot (10.4 mm / m) or 1 percent slope and shall be of approved corrosion-resistant material not smaller than the outlet size as required in Table 8-2 for air-cooling coils or condensing fuel-burning appliances, respectively. Condensate or wastewater shall not drain over a public way.

814.3 Point of Discharge. Air-conditioning condensate waste pipes shall connect indirectly to the drainage system through an airgap or airbreak to a properly trapped and vented receptors dry wells, leach pits, or the tailpiece of plumbing fixtures. Condensate waste shall not drain over a public way.

Chapter 2 Definitions

211.0 -I-

Indirect Waste Pipe - A pipe that does not connect directly with the drainage system but conveys liquid wastes by discharging into a plumbing fixture, interceptor, or receptacle that is directly connected to the drainage system.

222.0 -T-

Tailpiece - The pipe or tubing that connects the outlet of a plumbing fixture to a trap.

So you could have something like this, with the condensate pump.

If you can maintain the proper slope, you could remove the condensate pump.

In this situation the condensate pump is not required, since the tailpiece of the shower serves as the air gap. And you won't have problems with sewer gas, since the drain is protected by the trap on the shower drain.

You'll want to use proper fittings to connect to the tailpiece, rather than hacking together some leaking contraption. Without knowing the size and material of the condensate drain, I can't tell you exactly what fittings to use. In the end, you should end up with something like this.

You'll cut the tailpiece, and fit in a new 45° wye. Then you'll use whatever adapters and reducers are required, to go from the condensate drain into the PVC wye. The condensate drain has to maintain the minimum slope, so you'll have to work with that as a constraint.