There's no such thing as a free lunch

JavaScript (ES6), 88 85 bytes

Takes input as an array of strings. Returns 0 for not free or 1 for free.

a=>a.map(s=>([a,b]=s.split` `,t+={e:+a,c:-a,t:x=t*a/100,d:-x}[(b||'e')[0]]),t=0)|t<=0

How it works

Each line is split on the space to get a = amount, b = type of operation. If there's no operation at all (which is the case on the first line), b is set by default to "e" for "extra".

To add the correct amount to the total t, we use an object whose keys are the first letter of the operation:

{
  e: +a,           // extra
  c: -a,           // coupon
  t: t * a / 100,  // tip
  d: -t * a / 100  // discount
}

Note: If the bill consisted of only one element, map() would return a single-element array which would be coerced to an integer when applied the | operator, making the final test fail. But the OP confirmed that this can't happen. (Arrays of 2 or more elements are coerced to 0.)

Demo

let f =

a=>a.map(s=>([a,b]=s.split` `,t+={e:+a,c:-a,t:x=t*a/100,d:-x}[(b||'e')[0]]),t=0)|t<=0

console.log(f([
'12.34',
'15 tip',
'25 discount',
'1.5 extra',
'2 coupon'
]))

console.log(f([
'10',
'20 tip',
'20 discount',
'2 coupon',
'2 coupon',
'1 coupon',
'50 discount',
'2.55 coupon'
]))

Jelly,  42 39 bytes

⁾_@
⁾C×
”+
⁾‘×
ḲµṪḢO%7µĿṭ
ḢW;Ç€j”µFV>0¬

Takes a list of strings with decimal formatted numbers
(Leading zeros will work, but have the side effect of printing zeros to STDOUT prior to the final result).

Try it online! - not free; or free.

How?

⁾_@ - Link 1: a coupon
⁾_@ - literal "_@" - the Jelly code for subtraction with reversed arguments

⁾C× - Link 2: a discount
⁾C× - literal "C×" - the Jelly code for complement (1-input) then multiply

”+ - Link 3: extra cost
”+ - literal '+' - the Jelly code for add

⁾‘× - Link 4: a tip
⁾‘× - literal "‘×" - the Jelly code for increment (input+1) then multiply

ḲµṪḢO%7µĿṭ - Link 5, switch: char list
Ḳ          - split on spaces (gives [amount, type] as char lists)
 µ     µ   - monadic chain separation to get a value, say v
  Ṫ        - tail (get the type: "coupon", "discount", "extra", or "tip")
   Ḣ       - head (get the first character: 'c', 'd', 'e' or 't') 
    O      - cast to ordinal (99, 100, 101, or 116)
     %7    - mod 7 (1, 2, 3, or 4)
        Ŀ  - call link v as a monad
         ṭ - tack to the amount char list

ḢW;Ç€j”µFV>0¬ - Main link: list of strings (char lists)
Ḣ             - head - the base price char list
 W            - wrap in a list
   Ç€         - call the last link (5) as a monad for €ach of the rest
  ;           - concatenate
      ”µ      - literal 'µ' - Jelly's monadic chain separator
     j        - join all the parts with 'µ's             "10",".2 tip",".2 discount", "2 coupon","2 coupon","1 coupon",".5 discount","2.55 coupon":
        F     - flatten (makes a char list, for example: "10µ.20‘×µ.20C×µ2_@µ2_@µ1_@µ.50C×µ2.55_@")
         V    - evaluate as Jelly code (the above evaluates to -0.2499999999999991)
          >0  - greater than 0?
            ¬ - not

CJam, 45 42 bytes

q~Sf/(sd\{L(d\~ci6%"1\-* + )* -"S/=~}fL0>!

Take input as an array of strings, and takes the tip and discount as decimals.

Try it online!

Explanation

q~                e# Read and eval the input.
Sf/               e# Split each string by spaces.
(sd               e# Pull out the first element (base price) and cast it to a double.
\                 e# Bring the array back to the top.
{                 e# For each element L in the array:
 L                e#  Push L.
 (d               e#  Pop out the first element and cast it to a double.
 \~               e#  Bring the second element to the top of the stack.
 ci6%             e#  Mod its first character's ASCII value by 6. (c,d,e,t) -> (3,4,5,2)
 "1\-* + )* -"S/  e#  Push this string and split it on spaces.
 =                e#  Get the element given by number from the mod. CJam uses modular arrays,
                  e#    so 4 and 5 get elements 0 and 1 respectively.
 ~                e#  Eval whichever string was retrieved.
}fL               e# (end of loop)
0>!               e# Check if it's not greater than 0.

The code which is evaluated depending on the first letters:

t -> ")*"    Adds 1 to the tip amount and multiplies it by the current price.

d -> "1\-*"  Subtracts the discount amount from 1 and multiplies it by the current price.

e -> "+"     Adds the extra amount to the current price.

c -> "-"     Subtracts the coupon amount from the current price.