Hi,
1. In your payment methods settings, you have a setting "shipping methods". If you selected some shipping methods there, then the payment method won't appear on the checkout when another shipping method is selected.
In your case, you probably want to make sure that no shipping methods are selected in that setting of your payment methods.
2. Note that USPS shipping methods will only appear if the products in the cart have a weight and dimensions, and if the customer already entered his shipping address, that it is a real shipping address, and that USPS actually ships to that address normally with the shipping services selected in the USPS shipping method corresponding setting.
So that's a lot of conditions you need to check.
Regarding the weight approximation and dimension approximation settings, these allow you to adjust the size and weight of the packages so that it better reflects the reality of your packages. So for example if you have a product that weights 3 kg, the final package with the box itself, the wrapping, etc might weights 3.5 kg. And USPS only cares of the weight of the package itself, not the weight of the product inside. So you might need to enter a weight approximation of 120% for example to compensate for that difference. And for the dimensions, it goes both ways. The dimensions of the package might be bigger than the dimensions of the product. But if you have several products in the cart, the stacking algorithm will stack the products way less smartly than what you can do in real life, and thus the dimensions of the package might be smaller than what is calculated. So these options allows you to adjust a little. But if you leave these settings empty, then HikaShop will just use the weight and dimensions calculated "as is".
And for the rest of the settings below, you probably want to leave them as they are by default, unless you have some peculiar needs (like for example if your packages are not rectangular boxes).