Relationship between immunologic response to specific gluten epitopes and clinical antibody testing
1) Can a person be sensitive to barley hordeins, while being unaffected by wheat gluten or rye secalins?

Possibly, but I haven't found a documented case.

According to What is gluten? (Journal of Gastroenterology and Hepatology, 2017):


  A variety of sequences from α, γ, and ω gliadins, as well as from the
  glutenins, have been identified to activate coeliac disease. However,
  several hundred gluten peptides are predicted to be immunogenic and
  trigger the T-cell mediated response.6 The most immunodominant T-cell
  epitope is from α gliadin, although T-cell cross-reactivity against
  gluten-derived, secalin-derived, and hordein-derived peptides have
  been confirmed. Additionally, within each grain, there exists a
  distinct hierarchy of immunostimulatory gluten peptides.6 Of all the
  peptides known to stimulate the T-cell response in coeliac disease, a
  given patient may react to only a few.


However, according to Coeliac disease: immunogenicity studies of barley hordein and rye secalin‐derived peptides (International Journal of Experiemntal Pathology, 2016):


  In conclusion, barley hordein and rye secalin appear to contain CD
  toxic epitopes that are found in wheat gluten, as demonstrated by the
  observed cross‐reactivity of CD‐derived gluten‐sensitive small
  intestinal T‐cell lines to wheat gliadin, barley hordein and rye
  secalin‐derived peptides.


2) Are the blood tests specific to wheat gluten antibodies?

I can't say for deamidated gliadin peptide Ab IgG test, but tissue transglutaminase antibodies IgA test (anti-tTG IgA) is not specific for wheat gluten; antibodies can be triggered by wheat gliadins, barley hordeins and rye secalins. 

Gluten peptides (gliadins, hordeins, secalins) bind to tissue transglutaminase which deamidates them. Individuals with celiac disease have HLA DQ2/DQ8 dendritic cells, which present deamidated peptides to T cells, which stimulate B cells to produce antibodies against tissue transglutaminase: 


  In CD, deamidation of gluten by tissue transglutaminase (tTG) in the
  small-bowel lamina propria promotes presentation of gluten peptides
  (gliadin in wheat, secalin in rye and hordein in barley) by
  HLA-DQ2 or HLA-DQ8 dendritic cells to pathogenic local CD4+ T
  cells...The most widely used serological test is anti-tTG IgA
  (Medscape)


3) Can barley affect the gut differently than wheat? 

Possibly.

Nutrition and Celiac Disease (Nutrients, 2014):


  There is very limited data looking at the effect of barley hordein or
  rye secalin on CD outcomes in the published literature (e.g.,
  [66,67]), but evidence exists that these prolamins induce effects
  different to wheat gluten, at least at an immunologic level.


4) Do positive DGP and TTG test confirm celiac disease? 

The specificity of IgG deaminated gliadin peptide (DGP) is 98% and of IgA tissue transglutaminase (TTG) 95% (American Family Physician, 2014), so when both tests are positive, celiac disease is very likely. The final diagnosis is by histological examination of a tissue sample obtained by duodenal biopsy. 

5) Is it possible that, in celiac disease, symptoms disappear after removing only beer, but not wheat and rye from the diet? 

It could be possible to have celiac disease confirmed by blood tests without any symptoms despite consuming wheat, barley and rye. Beer could trigger symptoms by irritating the bowel, not by gluten, but by alcohol or other substances, like in people with irritable bowel syndrome (American Journal of Gastroenterology, 2013).