Splicing into factory stereo

If you're using the converter for a subwoofer amp then you should just strip off some insulation from the factory wire, twist the correct converter wire onto it, and solder.  wrap it up with some electrical tape.  Or you can cut the wires, strip insulation off the factory end, insert that into one end of a butt connector, and then twist together the end that goes directly to the spekaers with the end that goes to your LOC.  and then insert the two wires into the other end of the butt connector and crimp together.  When done right this will give you  apretty solid connection; but still I always prefer to solder onto the wire and leave it intact.

If however you are doing an aftermarket 4-channel amplifier (but still using the stock source unit and speaker wiring), then you'd want to cut the speaker wires instead of leaving them intact as you would with a sub amp install.  This way you can use the factory speaker wiring and hook it up to the 4-channel amp, but if u don't disconnect the wires from the radio side you could fry the radio when the 4-channel turns on.  Technically soldering is still the best method even if you have the end of the wire to work with.  However, you can also use butt connectors to crimp the wire connections instead.  I almost always use the small red ones (18-22 gauge I think) because they do a better crimp on the small factory wires than the larger blue ones (14-16 gauge). However, if you're doing the sub amp install that I first mentioned and are gonna be twisting together those wires you'll probably need one of the blue ones instead.

From doing research on the RCA line converters, I've noticed that the one I have in particular transfers up to 35 watts per channel.  So if I have a 200 watt amplifier, am I only using 35 watts of it(per channel)?  Or am I reading that wrong?

I originally had it installed at American in an Explorer Sport, which that might explain why my sub didn't sound that great afterwords.

Usually the power ratings that they include with some line output converters are what the converter can handle without causing the signal to distort at all.   Most of the time it will be an RMS spec. that's listed but sometimes they are unclear as to whether or not when they say "maximum power handling 35 watts" they mean peak power handling or the upper limit of the RMS power handling.

What kind of amplifier do you have?  Because what it says on the outside of it is most likely not anywhere near what it really puts out;  most manufacturers only display/advertise the peak power output even though it's basically meaningless.