[SalesForce] calling or using javascript function inside actionSupport

I think the user clicks on the Save button before the AJAX call is finished and the record is not saved, right? My solution is just to block the input elements on the page while the AJAX call works, and only if it finished - release the inputs. So user can not click on the Save button while your controller works.

Simple controller example:

public with sharing class test1 {
    // We will show some account on the page
    public List<Account> accounts { get; set; }

    public test1(){
        // Reading accounts
        accounts = [Select Id, Name, CustomerPriority__c From Account Limit 3];
    }

    public PageReference someAction(){
        // Your 'save' logic is here ....
        return null;
    }
}

Visualforce page. I will use JQuery to block the input elements on the page:

<apex:page controller="test1">
    <!-- Dynamically including the jQuery to the page -->
    <apex:includeScript value="https://ajax.googleapis.com/ajax/libs/jquery/1.9.1/jquery.min.js"/>

    <!-- This is a javascript function that will block and release elements -->
    <script>
        function blockAllInputs(param){
            if(param){
                jQuery('input[type=button], input[type=submit]').attr('disabled','disabled').removeClass('btn').addClass('btnDisabled');
                jQuery('select').attr('disabled','disabled');
            }
            else{
                jQuery('input[type=button], input[type=submit]').removeAttr('disabled').removeClass('btnDisabled').addClass('btn');
                jQuery('select').removeAttr('disabled');
            }
        }
    </script>

    <apex:form>
        <apex:pageBlock title="My Block">
            <apex:pageBlockButtons>
                <apex:commandButton value="Save" reRender="none"/>
            </apex:pageBlockButtons>
            <apex:pageBlockTable value="{!accounts}" var="a">
                <apex:column headerValue="{!$ObjectType.Account.fields.CustomerPriority__c.label}">
                    Value: 
                    <apex:actionRegion>
                        <apex:inputField value="{!a.CustomerPriority__c}" id="myField">
                            <apex:actionSupport event="onchange"
                                                action="{!someAction}"
                                                reRender="myField"
                                                onsubmit="blockAllInputs(true)"
                                                oncomplete="blockAllInputs(false)"/>
                        </apex:inputField>
                    </apex:actionRegion>
                </apex:column>            
            </apex:pageBlockTable>
        </apex:pageBlock>
    </apex:form>
</apex:page>

How it will looks like:

Try to use jQuery to trigger on the keyPress/keyDown events. Additionally i leave the value parameter empty if i want to set it from another javascript function and use reRender to refresh some areas:

<script>
jQuery('[id$=myField]').on('keydown keypress keyup', function(event) {
    if (event.keyCode == 13) {
        event.preventDefault();
        insertOffer(jQuery(this).val());
    }
});
</script>


<apex:actionFunction name="insertOffer" action="{!setNewBidOffer}" reRender="myField">
     <apex:param name="currOffer" value="" assignTo="{!currentOffer}"/>
</apex:actionFunction>

<apex:inputText value="{!currentOffer}" id="myField"/>

But one thing i don't understand: you trying to set the currentOffer variable to the currentOffer variable?