using simple autoencoder for feature selection
Autoencoders normally aren't linear models. If you make them linear (i.e. you create a shallow Autoencoder with linear activations) then you get exactly a PCA result. The power of Neural Networks is their non-linearity, if you want to stick with linearity go for PCA imho.

Keep a Train-Validation-Test set split, and try different configurations of hyperparams checking their performance on Validation data. Alternatively there are many libraries, such as hyperopt, that let you implement more sophisticated Bayesian hyperparameter searches, but unless you want to be published at a conference or win some competition it's a bit overkill. If you're still interested, the internet is plenty of tutorials like this one.