How can I hear (or at least detect) a pulsar at home?
Interesting questions - I believe I have answers to them.

You mentioned folding analysis - yes, you will need to do folding analysis to find pulsars - the majority of pulsars are found with what are called prepfold plots - here is some software that is very popular, but you can do this yourself. The general principle is that if you separate a signal at a time interval equal to the period of the pulsar and stack the resulting sections of the signal, you're effectively doing the equivalent of "image stacking" as you would do with images you take in a camera, which will boost the signal to noise ratio of the pulsar you're trying to look at.

The best bet for a small setup would be the Vela pulsar - here is a list of significant pulsars. The Vela is the brightest pulsar in the radio part of the EM spectrum (as per the link, which is from Wikipedia, so take its reliability with a grain of salt), so I would assume that's your best bet in trying to detect a pulsar.

As for antennas or dishes, I would think a dish would work best. You're going to want some decently sensitive equipment to detect a pulsar, so I would recommend a proper dish made for collecting radio waves.

And lastly, you would want to consider which frequencies the target pulsar emits at and check which ones your receiver can collect.

I think you've pretty much covered all the other issues; now you just need supplies and money :).