That is up for debate.
I really prefer if the current method stays: character rolls save against an effect-specific DC. Higher levels are already factored in through the increased ability scores, and the bounded accuracy means that low-level effects can be more easily overcome, but not so easily that they eventually fade out.
Yes, necessarily. I asked this on Twitter, and Mearls said that in a "target determines DC" system, the chance to resist would be the same.
Just because the designers are doing it doesn't mean it's necessary
However, I noticed what you wrote before, and now I think we've been arguing over 2 different issues... When you said "One is a 3rd-level novie. The other is a 20th-level archmage. The Fighter will have the same chance of averting both spells?", I was just thinking that DC doesn't have to depend on caster level (or spellcaster class level). In 3ed it didn't depend on that, and it wasn't a big deal*. Even in the case the DC would be always identical for both casters (but in general they are not, because of ability score increases), IMO it's not a big deal, because the higher level caster is still much more powerful. Then monsters abilities have their own DC, so the Dragon's DC is not going to be necessarily low in this system.
*basically I thought you were arguing that the same Cause Fear must automatically have a higher DC if cast by a higher level wizard, and I wanted to point out that IMHO that's not necessary; it CAN be done like that, but even if the game makes Cause Fear work identically for both wizards, the game is still fine
OTOH I know understand you were directly referring to the announced changes in the podcast, where DC depends on the target. I also don't like that!
In general, I don't like that much how in 5e every spellcaster has this level-based DC bonus, it's one more fiddly bit to keep track of. Then I expected they would introduce once again a 3e-style saving throws progression, which is 3 more fiddly bits (or maybe 6)... But DC based on the target is even worse, and as you say it might lead to high level monsters to be easier to save against, so how they will fix this? But of course, by giving each monster's ability a "this requires a save at -X", so once again fiddly bits!