Hi, I'm Daniel and I'll be helping you today I've been repairing white goods for 20+ yrs. and currently work for a well-know manufacturer.
Theres nothing to worry about that.
All this is is an anti vibration pad. They wrap it around the pipe work to take the vibration out of it when the compressor is pumping the gas around the system. But over time with the heat the compressor produces it starts to melt and looks a bit of a mes.
But don't worry about it nearly all fridges do this that have the vibration pad on them. It wont cause any problem to the appliance at all.
Yeap you can run these without a filter. They usually use an external filter so the fill hose goes from the water supply through the filter and in to the fridge. So if you don't have a filter you can just run the water pipe from the supply straight into the appliance itself.
If your water is clear and pure then no you don't need a filter. A filter is only used in lime scaled areas like what we have in parts of the UK.