Given a symbolic expression how to find if starts with a minus or not?

You can do it the way a human does it: Look for the "-" in the string.

#==1&/@StringCount[ToString/@InputForm/@negativeTruetests,Alternatives["-",Repeated["("]~~"-"]]

{True,True,True}

#==1&/@StringCount[ToString/@InputForm/@negativeFalseTests,Alternatives["-",Repeated["("]~~"-"]]

{False,False,False}

There are probably many creative ways to break this one horribly, but for the test cases it works:

(# /. Thread[Variables[#] -> 1]) < 0 & /@ negativeTruetests

(* {True, True, True, True, True, True} *)

(# /. Thread[Variables[#] -> 1]) < 0 & /@ negativeFalseTests

(* {False, False, False, False, False} *)

I will use your test lists alone as reference. If they are incomplete you will need to update the question.

When looking for a pattern for a group of expressions it is helpful to look at their TreeForm:

TreeForm /@ {negativeTruetests (*sic*), negativeFalseTests} // Column

You see that your True expressions always have the head Times with one negative leaf, be it -1, -2 or a Rational that is negative. Your False expressions either have head Times or Power but in the case of Times they do not have a negative leaf. Therefore for these expressions you may use:

p = _. _?Negative;

MatchQ[#, p] & /@ negativeTruetests
MatchQ[#, p] & /@ negativeFalseTests

{True, True, True, True, True, True}

{False, False, False, False, False}

Because of the Optional and OneIdentity⁽¹⁾ this pattern will also handle a negative singlet:

MatchQ[#, p] & /@ {-Pi, 7/22}

{True, False}

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Format-level pattern matching

Since it was revealed that this question relates to formatting it may be more appropriate to perform the test in that domain.

I will use a recursive pattern as I did for How to match expressions with a repeating pattern after converting to boxes with ToBoxes:

test = MatchQ[#, RowBox[{"-" | _?#0, __}]] & @ ToBoxes @ # &;

test /@ negativeTruetests
test /@ negativeFalseTests

{True, True, True, True, True, True}

{False, False, False, False, False}

Another approach is to convert to a StandardForm string and use StringMatchQ, which is essentially the same as the test above because StandardForm uses encoded Boxes:

test2 = StringMatchQ[ToString[#, StandardForm], ("\!" | "\(") ... ~~ "-" ~~ __] &;

test2 /@ negativeTruetests
test2 /@ negativeFalseTests

{True, True, True, True, True, True}

{False, False, False, False, False}