First, it would have been much better if this was posted in the "AWS (welding)" forum.

1.  D1.1 does not give a minimum spacing.  As discussed many times in the past year in these fora, a prudent welding engineer will not allow the Heat-Affected Zones to overlap, or even touch.

2.  Assuming that the D1.1-specified minimum preheat has been maintained throughout the welding, D1.1 does not require PWHT.  The Engineer-of-Record may require PWHT if he/she deems that the residual stress from welding will hurt the structure.  This is typically caused by environments that cause Stress-Corrosion Cracking in carbon steel, like caustic - NaOH and KOH - solutions.

"... not allow the Heat-Affected Zones to overlap, or even touch " brings up a question.
Say you have a WF column requiring stiffeners opposite the flange of a direct welded WF beam moment connection.  The stiffener plate full pene. welds to the inside of the column flange in the shop, the beam flange full pene welds to the outside of the column flange in the field --- typical moment connection, happens everyday in steel framed buildings. These two opposing welds, may be only 1/2" apart depending column size, will most likely create overlapping HAZ's.  How can this be avoided?  And is it really a problem?  I've never had a problem with it but have often wondered.  Thoughts?  
 

One half-inch is a lot when discussing HAZ's.  Instead of reading all the hogwash I've seen published in training texts, just "ask a weld".  Any D1.1 Welder Qual full [1-inch] thickness plate test that was 'bent' as opposed to x-rayed will do it.  Look at the bend 'straps', you will be able to easily discern the weld and the basemetal.  There will be a line about 1/16" wide between them -- the HAZ.  If you can't see it, lightly etch the strap with any mineral acid - muratic, sulfuric, HCl, etc.  I have even used battery acid from an Ironworker Forman's truck.  Battery acid takes a while, and 2-4 applications.  

Point is, despite a lot of otherwise reputable literature, HAZ's tend to be very narrow when compaired to the size of the weldmetal deposit.

Depending on thickness our practice is 50mm, to avoid the overlaping of the adjacent heat affected zones.