The answer to this puzzle is:
BACTEREMIA
My firsts signify education
My seconds axial examination
My thirds medical location
My fourths denote radiation
My fifths state abbreviation
Splitting this 10-letter word into 5 pairs of letters, and thinking of these pairs as 'firsts', 'seconds', 'thirds, 'fourths' and 'fifths', we have:
Firsts = BA - Bachelor of Arts degree (signifying education);
Seconds = CT - Computed Tomography, a 3D axial scanning technique (axial examination);
Thirds = ER - Emergency Room (medical location);
Fourths = EM - Electro-Magnetic (radiation);
Fifths = IA - the state abbreviation for Iowa.
My whole is blood contamination
My parts each two in variation
My consonants lack reiteration
Bacteremia is the contamination of blood with bacteria (also described by the title, which sounds like an immune response to infection). My parts each two in variation merely confirms that the clues to the first part above relate to two-letter sequences. Note also that the five consonants of the word do not repeat (lack reiteration).
When I am shaped like my first
Rows from my parts that may burst
Now comes an ASCII part, similar to another recent riddle of the OP's! Shape the letters of the word 'BACTEREMIA' like 'its first', i.e. a letter B:
BA
C T
ER
E M
IA
Adding flipped to in empty spaces
My second column through blood races
Adding the letters of 'to', but flipped (i.e. in reverse order) gives:
BA
COT
ER
ETM
IA
By reading down the second column you can read the word AORTA - the major blood vessel through which blood races.
Sinking first precedes one year
After it sympathize with fear
New second across for relaxation
And fourth row now for documentation
Finally trademarked my column third
These lines refer to other 'words' we can read in the diagram:
Sinking first = first column downwards: B CEE I, which can be interpreted as BC ONE - the year which preceded AD 1 in the calendar (precedes one year).
Second across = second row left to right: COT - a bed (for relaxation).
Fourth row now = fourth row left to right: ETM - Electronic Technical Manual (documentation).
My column third = third column downwards: TM - the abbreviation for trademarked.
Except my seconds, none a full word
Note that the only ones of these which are real words rather than abbreviations or acronyms are those in the second row and second column, as described here.
And thus it is solved!
