Don’t go cheap on the corks. They really are not that expensive anyways. I like 1 and 3/4 inch premium corks. I get those because you never really know how long you want to keep a bottle. A cork like that will last seven years. I have never tried the synthetic corks or the twist tops. It’s on the list.
Glass. Glass unless for some reason you cannot control the weight. Then by all means use plastic. I knew a guy who used plastic car boys and won first prize for his home craft beer.
The kind your doctor uses.
You will want a floor corker. Less stress and they can get the tough jobs done.