What do you mean by "removes other effects"? Do you mean that other effects disappear from the Effect Controls panel? Do you mean that you just can't see the results of the effects in the Composition panel?
I'll assume that you mean that latter.
The Timewarp effect operates on the original layer, not on the output of the effects that precede it in the effect stack. To apply the Timewarp effect to a layer that has other effects already applied, precompose the layer with the other effects first.
Since you're new to After Effects, I very strongly recommend that you start here and work your way through the basic materials before taking on a complex project:
I tried to pre-compose the layer with the other effects and then add the new composition to my project. After that, I applied the 'timewarp' effect and once again my cineon converter effect result wasn't visible in the RAM preview or the composition preview panel on top. So:
1. I removed the time-warp and let the other effects remain (Cineon Converter + Fadein/fadeout frames + Solid composite)
2. I went to layer and pre-composed as you said.
3. Added the new composition that got created and then added the timewarp effect to it.
4. Same result, it won't show my cinceon converter effect.