Are there bacteria that respire anaerobically in aerobic conditions?
Are there any organisms that would still perform anaerobic respiration even when shifted to aerobic conditions?


Yes, there are. They are called aerotolerant anaerobes as shown in this figure on tube 5. In the thioglycollate broth, growth is seen throughout the length of the tube

Legend 


Obligate aerobe
Obligate anaerobe
Facultative anaerobe
Microaerophilic
Aerotolerant anaerobes



  Are there any which can survive?


Obligate anaerobes wouldn't survive in aerobic conditions. Hence they are named so.


  Such a bacterium would be read false positive for oxidizers in such a case because it would ferment glucose even in aerobic conditions.


The thioglycollate broth shown here is used to differentiate bacteria into the above said criteria. The Hugh-Liefson test does'nt do that. You can not conclude that a bacteria is aerobic or anaerobic using that test. The H-L test, is interpreted in two ways.


Fermentative - When both the closed tube and open tube show a colour change
Oxidative - Colour change only in the open tube


In short, the words aerobic/anaerobic are not used to interpret the H-L test. But we can draw parallels.


Obligate aerobe - Open tube (+), Closed tube (-) - Oxidative
Obligate anaerobe- Open tube (+ at bottom), Closed tube (+) - Fermentative
Facultative anaerobe- Open tube (+), Closed tube (+) - Fermentative
Microaerophilic- Open tube (+), Closed tube (-) - Oxidative
Aerotolerant anaerobes- Open tube (+), Closed tube (+) - Fermentative