Knowing this is 64-bit SQL Server 2008 Enterprise simplifies the discussion, so thanks for that. The first thing to say is that it is almost never right to leave the max server memory configuration option at the default of 2147483647.
The max server memory option only limits the size of the buffer pool (used for all single-8KB allocations pre-2012). SQL Server can also require significant amounts of non-buffer pool memory (allocations > 8KB). Other instances, applications, and the OS also need room to work (and 2GB is not always a good rule of thumb there). Amit Banerjee, an Escalation Engineer with Microsoft PSS has three excellent posts on this topic, which I encourage you to read.
You will generally want to set max server memory to a fairly conservative maximum value to start with, monitor the actual amount of free memory on the server over a full load cycle, preferably over a decent period of time, and gradually increase it as appropriate and necessary.
The exact level to start at depends on how much non-buffer pool memory SQL Server might be expected to require, and how much memory will be needed by other instances, applications and the OS. On a server dedicated to a single instance of SQL Server, I might start max server memory around 160GB with 192GB installed. Your circumstances might dictate that a much lower starting point is indicated.
I also recommend you consider using the Lock Pages in Memory (LPIM) option (that link explains why in great detail). This will be enabled by default if the SQL Server service is running as LocalSystem (which is not a best practice). See this CSS article for details including how to tell if SQL Server is currently using LPIM or not.
Memory used via this mechanism is not visible to Task Manager; it is better to monitor SQL Server memory usage via Performance Monitor counetrs and/or the sys.dm_os_sys_memory DMV.
You say you enabled AWE on SQL Server but did you activate /PAE in the boot.ini file?
PAE enables Large Memory support.
Activate PAE in Windows 32Bit Enterprise
c:\boot.ini
[boot loader]
timeout=30
default=multi(0)disk(2)rdisk(0)partition(1)\WINDOWS
[operating systems]
multi(0)disk(0)rdisk(0)partition(2)\WINDOWS="Windows Server 2003, Enterprise" /fastdetect /PAE
http://support.microsoft.com/kb/283037
Lock Pages in Memory
As part of enabling AWE, you would also need to give the SQL Server Service user the right to Lock Pages in Memory. Enabling this option in the Group Policy dialog (gpedit.msc), prevents Windows from paging memory out of the SQL Server working set for its own needs, especially if memory starts to run low on Windows.
http://msdn.microsoft.com/en-us/library/ms190730.aspx
Monitoring AWE Memory Usage
To test AWE usage, I setup a Windows 2003 Enterprise SP2 with PAE enabled in boot.ini. I used SQL Server 2005 32bit Standard SP4.
The total physical memory is 8192Mb. The max memory setting for SQL Server is configured at 8192 and the minimum is at 2048.
Does SQL Server see AWE on startup.
2012-02-09 07:27:51.35 Server Address Windowing Extensions is
enabled. This is an informational message only; no user action is
required.
Is SQL Server using Locked Pages for the Buffer Pool
According to some blogs, I should be seeing this message, but I'm not.
Using locked pages for buffer pool.
This is an informational message only; no user action is required.
Memory being allocated through AWE
SELECT SUM(awe_allocated_kb) / 1024 AS awe_allocated_mb
FROM sys.dm_os_memory_clerks ;
[3648]
Memory allocated to Multi-Page memory.
select sum(multi_pages_kb)/1024 as [MultiPage Memory, MB] from sys.dm_os_memory_clerks
[14]
Multi-page memory can not use AWE allocated memory
Memory being used outside the buffer pool
SELECT sum(multi_pages_kb
+ virtual_memory_committed_kb + shared_memory_committed_kb) AS
[Memory used outside BPool, mb]
FROM sys.dm_os_memory_clerks
WHERE type <> 'MEMORYCLERK_SQLBUFFERPOOL'
[24]
Overall memory allocation by component
SELECT type, (single_pages_kb)/1024 as Single_Pages_MB, (multi_pages_kb)/1024
AS Multi_Pages_MB, (awe_allocated_kb)/1024 as AWE_allocated_MB
FROM sys.dm_os_memory_clerks
GROUP BY type
ORDER BY 2 DESC
This will show you how single-page and multi-page memory is being used by each component. It will also tell you which one is able to benefit from AWE.
Mutli-page : when request exceeds 8Kb
Single-page : when request is less than or equal to 8Kb
As a side note, I have found sp_whoisactive very helpful in finding slow queries, memory hogs and just about everything that's going on in SQL Server. Here is a link that Brent Ozar provides on how to set it up and use it.
http://www.brentozar.com/archive/tag/sp_whoisactive/
Best Answer
I will answer the last question first: Yes, you can change it while the server is running without issue. If you want to change the value via SQL you can do it with the following query
See this page for more details about setting memory on SQL server.
Your first question, unfortunately the answer is: I can't tell you, I'm not there.
There is a 1,000,000 things you need to factor in when allocating memory. How big are result sets from the queries, how often are they run, would a query that used to take 20 ms be ok to now take 200 ms?
Sql's defaults assumes that it is the only thing running on the server, so it just sets the memory to MAX_VALUE and it stops growing when all available memory is in use (and on dedicated hardware that is fairly close to what you want to happen(see Aarons comment for a possible caveat)). Normally any web server or other software interacting with the database would be on different hardware communicating to it over the network.
You really just need to just set it to a value you think is sane, and if your webserver is still memory choked lower it. If SQL is not giving you enough performance after you give the webserver the memory it needs you will need to either buy more ram or move the SQL to dedicated hardware.