I agree that in your situation no amplifier should be needed. In fact, an amplifier would be at risk of overloading.
To answer your question,
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4) If an amplifier is necessary, can it go at the end of the run just before the receiver? Or do I need to get it closer to the antenna?
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An amplifier is best located upstream of the loss(es) that it's to overcome. The closer an amplifier is to the antenna, the more coax and splitter loss will be connected to the output of the amplifier... the amplifier can then be effective at overcoming those losses. Amplifiers can not 'pull' signal from an antenna, out of the air or out of the coax. Amplifiers can only 'push' signal through loss on the output side of the amplifier.
If installed at the end of the run, you can only hope that an amplifier would have less noise than the tuner... if this is true, you would get a small benefit, if not, the amplifier may actually add more noise to the total system, making reception worse.