Electronic – Losses in NE555 oscillator and Sallen-Key filter

To amplify the signal first is debatable. In general, a beefier unfiltered signal may be easier to filter carefully than a weaker one amplified after the fact. But this is not a fundamental rule.

What you need to think about most are noise levels and impedances. When dealing with weak signals, you'd be wise to never do anything to reduce the signal-to-noise ratio. Impedance matching at the desired frequency of 800kHz maximizes this. But you probably won't need to be super-precise or fancy about it.

If your desired signal is lost in a strong, messy broad-spectrum signal, an amplifier could saturate or have other troubles. Whether that is the case or not, you probably want at least a simple passive filter in front of the amplifier, such RC low-pass and high-pass filters, to reduce excessive noise of frequencies far from your desired 800kHz signal. Amplify the result, then send it into a more precise bandpass filter.

Can amplifying and filtering be done with the same op amp? Sure, but you have to work the numbers and know what the chip is capable of doing. Your question doesn't say how sharp the filter needs to be - what the width of the bandpass should be, or if it's important to have a very flat passband, or if pulse shapes or delay characteristics are important. If any of that matters, then the filter circuit may have to sacrifice gain and the more likely you'll need a separate amp.

One ting that jumps out is you have not selected a model for the diode - right click and select one from the list. Otherwise it may not simulate correctly.

I think what you are seeing may be to do with the rectifier not being able to sink current (only a small amount will flow through the 200k resistor), so the input signal will not look as it would unloaded, it will "charge" to it's peak. Then the larger the cap, the longer the fall time when the signal stops. Try putting a non-inverting buffer in between the rectifier and Sallen Key input. This should make both filters respond in the same way.