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:
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
It looks like you're new here. If you want to get involved, click one of these buttons!