If I burn onto a blank DVD-R that I bought in US, will it play in both European and US DVD players?
Yes - as long as you make a region free dvd copy. Most (european, as AKED pointed below) dvd players can automatically read Pal or NTSC, case that your content it's not progressive and/or 1:1
Do I need to get disks that are of a certain Region?
No - just burn copy region free.
In my experience most dvd reading problems come from encoding content with too high a bit rate.
That's the reason many of us used toast and other similar "user friendly" soft to burn DVD discs (with a decent bit rate) IF we wanted to assure maximum compatibility throughout players... that was some years ago, honestly dunno how are things now
The blank DVDs are neutral;, no region. Dvd ripping programs should have a menu to strip the dvd of region. Select "0" as region. Also you can hack your DVD or Bluray player to be region free if you get a DVD with another region:
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