Op Amp Voltage and Current Output
Your circuit has some major flaws.
You have correctly calculated the R1, R2 voltage divider, but it gives 0.396 volts above ground. If you swap R1 for R2, it will give the desired 5.6 volt.
The solar cell supply voltage of 6v is actually a variable supply - with no light source, it will sag - your circuit should work properly for any input voltage. Since your 5.6v reference voltage is derived from this supply, it will sag too. When it falls below your battery voltage, the op-amp will discharge the battery - certainly not what you want.
Most op-amps include internal current-limiting circuits. Many common op-amps limit current to about 0.03 amps. So the answer to your Question 2 is no, charging current will be less than 0.2 amps. Furthermore, even 0.03 can heat the op-amp considerably.
Addressing your question 1: Many op-amps require an internal voltage drop before charging current is allowed to flow. Some op-amps brag about "rail-to-rail" output voltage. These have much smaller voltage drop. The term "rail" refers to the op-amp supply pins (V+ and gnd).
Be not discouraged. What you are attempting might seem straightforward. Observe how complex the internal circuitry is inside a battery-charging integrated circuit: