Python3, 458 bytes

E=enumerate
def f(b):
 q,s=[(b,0)],[b]
 while q:
  q=sorted(q,key=lambda x:len(x[0]))
  b,c=q.pop(0)
  if len(b)==1:return 1
  v=[x for i,a in E(b[:-1])for x in[i,i+1]if -a==b[i+1]]
  Q=[([a for i,a in E(b)if i not in v],0),(b[1:]+b[:1],c),([[i,[1,-1][i==1]][i==1]for i in b],c),([{1:2,2:1,-1:-2,-2:-1}[i]for i in b],c)]
  if 0==c:Q+=[([j for i in b for j in[[[-2,-1],[1,2]][i==1],[i]][abs(i)!=1]],1)]
  for X,Y in Q:
   if X and X not in s:q+=[(X,Y)];s+=[X]

Try it online!

Charcoal, 97 bytes

⊞υθＦυＦ…±²¦³Ｆ∧κ÷⟦²±²⟧κ«≔⟦⟧ηＦι≡μκ⊞⊞Ｏηκ±λ±κ⊞⊞Ｏηλ±κ⊞ημＷ⌊Φη⁼ν±§η⊕ξ≔Φη∨⁻↔μ↔ν⁺ν§η⎇⁼μν⊕ξ⊖ξηＰ⁼¹Ｌη¿‹ＬηＬι⊞υη

Try it online! Link is to verbose version of code. Outputs a Charcoal boolean, i.e. - for primitive, nothing if not. Explanation: Performs an exhaustive greedy search, which the hints suggest is a valid approach.

⊞υθＦυ

Perform a search for moves that reduce the length of the list.

Ｆ…±²¦³Ｆ∧κ÷⟦²±²⟧κ«

Applying moves 5. 2. and 1. might reduce the length of the list if there are enough occurrences of 1, -2 and/or 2, -1, but there are other options via moves 3. and 4. as listed below.

≔⟦⟧ηＦι≡μκ⊞⊞Ｏηκ±λ±κ⊞⊞Ｏηλ±κ⊞ημ

Apply move 5. (and moves 3. and 4. where applicable) to the list.

Ｗ⌊Φη⁼ν±§η⊕ξ≔Φη∨⁻↔μ↔ν⁺ν§η⎇⁼μν⊕ξ⊖ξη

Apply moves 2. and 1. to delete all cyclically adjacent negations. (Normally these are only the ones introduced by move 5. but on the first pass there may be some bonus adjacencies in the input; applying move 5. doesn't affect them since e.g. 1, -1 becomes 1, 2, -2, -1 and so the original 1, -1 will collapse anyway. As this is code-golf I put off the cyclic reduction until this point as I can use it to clean up after move 5. as well.)

Ｐ⁼¹Ｌη

Note whether this identifies the original list as being primitive.

¿‹ＬηＬι⊞υη

If this resulted in a shorter list then push it to the search list. (This only applies one of the relevant hints but it would be less golfy to apply the other one.)

Rule combinations and their effects:

• 5.2.1. Tries to delete 2 before -1 or -2 after 1.
• 3.5.2.1.3. Tries to delete 2 before 1 or -2 after -1.
• 4.5.2.1.4. Tries to delete 1 before -2 or -1 after 2.
• 4.3.5.2.1.3.4. Tries to delete 1 before 2 or -1 after -2.
• 3.4.5.2.1.4.3. Tries to delete -1 before -2 or 1 after 2.
• 3.4.3.5.2.1.3.4.3. Tries to delete -1 before 2 or 1 after -2.
• 4.3.4.5.2.1.4.3.4. Tries to delete -2 before -1 or 2 after 1.
• 4.3.4.3.5.2.1.3.4.3.4. Tries to delete -2 before 1 or 2 after -1. (Same as 3.4.3.4.5.2.1.4.3.4.3.)