1) A thermistor reduces the voltage drop. You need an NTC thermistor, so that the resistance is largest at turn on, and then the thermistor heats up and its resistance decreases, so there is less voltage drop.
It depends on your load, and how sensitive it is to supply voltage, as to whether a thermistor would give you any improvement at all over a simple resistor.
2) The 100 Ω series resistance was a guess. To get the same voltage drop with 10 lamps you will need 10 Ω, and that could well be too small to protect the relay. You still haven't said what the power rating of the lamp is and without that I don't know what resistance it could stand. You would be able to use more lamps with an NTC thermistor than with a resistor.
The peak current with a 100 Ω resistor and a 240 V supply is about 2.4 A. However if the lamps are 15 W, and you have 10 of them, there will be a significant voltage drop with a 100 Ω resistor.
3) The relay is failing because you are massively overloading it. When you connect a capacitor to a supply, it charges extremely quickly. The current is only limited by the resistance of the wires. You could easily be getting 100 A peak current, which is far more than he relay can handle, and it occurs as the contacts close, so there isn't much contact pressure. You are simply welding the contacts together.
Relays are fairly robust, and you don't have to keep the current spikes to less than 5 A for a 5 A relay, but it is a good idea to keep the peak current to no more than a few times the continuous rating.
I have had the same problem with a PIR light switch turning on a fluorescent lamp that contained a PFC capacitor. A thermistor stopped it happening again. There is also an issue turning on low voltage halogen lamps as they can take massive currents when the filament is cold.