The tuner adapter(TA) doesn't "require" a splitter, but you can use one with it. You can either go coaxial > splitter > 1 to TA input and 1 to Tivo OR coaxial > TA input > TA output > Tivo. The TA has a splitter inside and a small passive amp that helps make up for the signal loss of the split. Unfortunately this also slightly lowers the DS SNR and increases the Tx like any amp would. In general, if you have stronger signal, it betters to go through a splitter, but if you signal is on the weaker side, it's better to go through the TA.
Also, pixelation is not caused by a issue with the TA. The TA only sends the signal to Cox to enable certain channels. It's a on/off type of system, so if you get the channels, the TA is working. Pixelation would be a sign of a issue with the downstream signal or the tuner/cable card inside the Tivo. The Tivo has diagnostics you can use to see the signal levels. A amplifier specifically with a active return wouldn't help, because the return, or upstream(Tx) has nothing to do with picture quality. However if the Tivo is reporting low Rx(IB) then a passive amp "may" help, but it would depend on the specifics.