What converter to use?
Pins to another board (like a "perf board") with pads (or a PCB). Same as #2.
"Safe" is relative for those things. None of them bear any approval markings, so you have to assume that each could kill you or burn your house down. If you have no datasheet there's no way to tell how inefficient they are. They're probably just as (un)safe in a moderate size enclosure as a huge one. 
That one looks a bit better designed (EMI filter, inrush NTC) but not everything that could kill you is visible.
That one might have non-fake approvals but as you say is not convenient. 


If this is a real application, I strongly suggest you use something from a reputable manufacturer with (non-fake) approvals. It will cost more, but you'll be able to sleep at night. Try a small (eg. 10W) enclosed supply from Meanwell (from a reputable franchised distributor not ebay/Ali).

All your questions should be answered by the manufacturer's data sheets and supporting documents (application notes etc.). If they can't supply those documents, you should not buy the product, in general but particularly for a safety-critical component.