How does TP4056 handle the charging process it more than 5W is provided to the module?
The 5 W isn't provided to the TP4056. There should be 5 V available and the maximum current that the TP4056 will draw is 1 A. 5V x 1 A = 5 W. It is the TP4056 that determines how much current is drawn so that means that it also determines how much power is drawn (because power = voltage x current). The supply should provide 5 V and cannot influence the current.
Will the battery stop charging or will it damage the cell?
The TP4056 decides when the battery is full, that is when the battery has reached a voltage of 4.2 V (and the charge current drops below a certain value). Then the TP4056 simply stops feeding current into the battery. Batteries cannot "stop charging", it you apply enouhg current/voltage they will keep charging and do so until they self-destruct.
Also how much more than 5w it will keep charging the batteries?
Your assumption it wrong, the charging stops so no power is fed to the battery.
If I provide more power than 5W but the current is less than 1A but the voltage is higher making the power more than 5W then what will happen?
You should not be providing a certain amount of power to the TP4056, you should supply a voltage. That voltage needs to be 5 V. The power that the 5 V supply can deliver should be 5 W or more. A 5 V, 1 A supply is OK but a 5 V, 10 A supply (that is a 50 W supply) is also OK. The TP4056 will only draw up to 5 W even though the supply can deliver 50 W.
Will it charge the cell properly?
Yes, as long as you supply 5 V.
If I provide more power to the TP4056 so will it be possible to draw power for load from the output point of the module?
You should not be providing more power and you cannot, you should provide 5 V.
If I attach 2 wire at both positive and negative terminals of the cell then draw power from one of the circuits and use another for charging so when the battery will start charging will it automatically stop discharging or there will be another case?
No, the TP4056 is not suitable to charge a battery while there is also a load taking current connected to the battery. Then the current will never drop below a certain value so the TP4056 simply keeps charging for eternity. The TP4056 is a simple charging IC. Only use it to charge a battery that is not connected to a load.
I suggest that you take a step back and first study and fully understand how things work with power supplies and a load regarding voltage, current and power. There are many questions with answers to be found on this site. Read them!