shipment modules
There are stock shipping modules, which I highly recommend (https://docs.zen-cart.com/user/shipping/shipping/). Or you can add things like UPS, USPS etc, however, then you play by their rules & APIs. So if UPS is responding slowly with quotes for example, people cannot checkout.

payment modules
The easiest by far is Square, which is built in =), as well as PayPal.

understanding how weight factors in final order total
This is complicated because all of the carriers use DIM weight, not raw weight like Zen Cart. Generally I urge folks to fashion a table, preferably with a free shipping level. You will never be right on the money no matter what you choose. Also look in to tare settings (https://docs.zen-cart.com/user/shipp...ulations/#tare). Test, check, test and check some more.

how to test zencart email (probably by a transaction to myself).
Yep, to more than one email address if possible. Also, make sure to set up SMTP to lower the spam score of these emails (https://docs.zen-cart.com/user/email..._smtp_servers/).

~Melanie