using a capacitor for an off-grid system
The quick answer is no. A capacitor won't help, AND you don't need it. If you have a 400 - 800 AHr battery, it will provide you with all the juice you need for a small appliance such as you've specified. Additionally, without going into details, practical capacitors simply don't have the energy and power density you want. 

If you're worried about transient load currents (as you should be), you need to pay attention instead to your inverter specifications. Inverters don't generally have much excess margin on their load current specs, so you need to get one which seems oversized compared to your average load. The exact amount of this overage will depend on exactly what appliances you're using.

Recharging via exercise bike won't hurt, but it won't help a whole lot, either. The general rule of thumb for humans doing work (like driving a generator) is 100 watts. This is (roughly) the solar power incident at noon on a 1 square foot solar cell. Figure 20% efficiency for the cell, and a 1-hour workout on the bike is more or less the equivalent of an extra square foot of solar cell over the course of a bright day. Like I say, it won't hurt, but its major impact will be to make you feel better about exercising and not wasting energy.

By the same token, running appliances for short periods of time need to be taken into account, but in general they are not dominant in your energy budget. Let's say a blender draws 10 amps at 120 volts. For a really good inverter with a 12-volt input, that's an inverter draw of 100 amps. But, it only lasts for 1 minute, so the total battery drain is 100 amps x 1/60 hours, or about 1.5 amp-hours.