Retina, 35 31 29 bytes

Saved 4 bytes by taking some inspiration from ETHproductions.

Thanks to Leo for saving another 2 bytes.

\d+|\+
$*
1+
$.&
1`!

!\d*!?

Try it online!

JavaScript (ES6), 58 56 bytes

Saved two bytes thanks to Martin Ender.

let f =
x=>x.replace(/[^!]+/g,eval).replace(/!(\d*)!?\d*/g,"$1")

<input value="2+2!5+5" oninput="try{O.value=f(value)}catch(e){}"><br>
<input id=O value="410" disabled>

Might be improved somehow...

Jelly, 18 bytes

ṣ”!µḊm2;@ḢW$LÐfVṾ€

Try it online!

How?

ṣ”!µḊm2;@ḢW$LÐfVṾ€ - Main link: string
ṣ”!                - split on '!' characters
   µ               - monadic chain separation (call that x)      e.g. ['1+1','0+0','0+0','0+0','','1+0','','','']
    Ḋ              - dequeue x (all but the leftmost entry of x) e.g.       ['0+0','0+0','0+0','','1+0','','','']
     m2            - modulo 2 index into that result             e.g.       ['0+0',      '0+0',   '1+0',   '']
           $       - last two links as a monad
         Ḣ         -     head x (the leftmost entry of x)        e.g.  '1+1'
          W        -     wrap                                    e.g. ['1+1']
       ;@          - concatenate with reversed arguments         e.g. ['1+1','0+0',      '0+0',   '1+0',   '']
             Ðf    - filter keep:
            L      -     length (keep that have non-zero length) e.g. ['1+1','0+0',      '0+0',   '1+0']
               V   - eval as jelly code (vectorises)             e.g. [  2,    0,          0,       1]
                      Yes, addition is just + and decimal numbers are just strings of digits in Jelly believe it or not!
                Ṿ€ - uneval €ach (creates a string from each one)e.g. [ '2',  '0',        '0'     ,'1']
                      without the € it would uneval the list and hence yield commas too)
                   - implicit print (prints the resulting list [of characters and possibly
                      lists of characters] as if it were all one string.)

CJam, 20 bytes

r'!/{_{'+/1b}&}%(\2%

Try it online! (Linefeed-separated test suite.)

Jelly, 16 bytes

ṣ”!µḢW;m2$VṾ$L¡€

Try it online!

Explanation

The key observation here is that we can run the steps "out of order"; instead of evaluating the sums then ignoring the ones we don't like, we can ignore the sums in invalid positions, then evaluate the rest.

ṣ”!µḢW;m2$VṾ$L¡€
ṣ”!                Split input on '!'
   µ               Set as the new default for missing arguments
    Ḣ              Take the first element, removing it from the default
     W;  $         Cons with
       m2            every odd-numbered element of {the tail of the !-split input}
               €   For each remaining element
          VṾ$      Evaluate and de-evaluate it
             L¡      a number of times equal to its length

Evaluating a sum like "10+10" will evaluate it to a number, e.g. 20, then de-evaluate it to a string, "20". Repeating that process has no additional effect (it's idempotent). Thus, we effectively evaluate every element of the string, except the null string, which remains unevaluated because it has a zero length.

Batch, 192 184 bytes

@echo off
set/ps=
call:c . "%s:!=" "%"
echo(%s%
exit/b
:c
set s=
if not %2=="" set/as=%2
:l
shift
shift
if "%1"=="" exit/b
if %1=="" goto l
set/at=%1
set s=%s%%t%
goto l

Having to handle the empty strings is inconvenient.

Ruby, 58+1 = 59 bytes

Uses the -p flag.

i=0;$_=' '+$_;gsub(/!?([^!]*)/){(eval$1)if(2>i+=1)||i%2<1}

Try it online! (An extra line of code was added so that it would take all of the input lines and print the output in different lines.)