I've never tested it but would expect that the answer to both questions is a big 'it depends'.
#1 is far more likely to be 'yes' but it will be display dependent. That is, many new displays, since it they all use digital processing, don't have a huge issue accepting a PAL format and converting it to properly display on the screen. But, I don't think at the core they change the refresh rate from 60hz (NTSC) to 50hz (PAL). I think they just convert the signal to something that the display is happy with.
So, the real answer to your specific question is no. ALL TVs won't do it, some will and you will likely see specifications for those TVs that say PAL compliant somewhere for signal compatibility.
#2 is more likely a 'no' I would believe instead of a yes. That is, the DVD recorder FIRST must be PAL compliant (not as likely) and then would likely just record in the PAL format to disc. There is no necessity to convert from PAL to NTSC for the recorder to work, and the burden is on the TV to go from PAL to NTSC, not on the player. Better DVD recorders designed for world use, likely will do this processing on board and you will be able to specify output, but this is not something I would ever consider 'standard'. Many DVD players DO have the ability to read PAL discs and convert them to NTSC quite easily. This is not the same as passing a signal through the DVD player though. So, once again, player dependent.