Do all iPads require monthly payments?

the iPad can connect via wifi to your existing router ... what you're seeing is that the higher value iPads all have cell service embedded and their price is covertly subsidized by the cell provider.  Yes, you can buy it without that feature enabled -- and without the subsidy.  You can also check with your current cell service provider and see if they'll come up with a deal on one for you ... because the implied threat is that you'll quit their service and go with the competition to get the 'cheaper' iPad price.  -- grampa

No.

If you however decide to get one that has a SIM slot, the SIM will require connection to a phone company which you will have to pay to get access from. Payment plans can be post-paid (usually monthly) or pre-paid (as needed). Even if your iPad has a SIM option, if you don't use it, you don't need to pay to the phone company. You can still connect through your home's wifi, if you already have one. 

All iPads are wifi capable. This means that you can connect to the Internet via your home wifi; the iPad will use whatever provider you use for your computer etc. If you want to get an iPad that is cellular capable, that means you will have to have an account through a cell provider and that will cost you extra.

not if you pay the full cost in a cash lumpsum upfront