Problem with Reverb/echo

Two things come to mind:

1) You simply have latency too high. That causes you to hear the incoming sound a plit second late. It's does sound more like a delay, though, not like a reverb. You set latency/buffers/drivers in the Options dialogue.
With your built in soundcard you can't expect very good results, but if you got it working with your previous PC, then it should work equally well, assuming the machine is equally capable.
With your onboard card it's likely you can only use either MME 32 drivers or ASIO4ALL (free download).

2) Depending on where you get the monitoring signal (via the mixer or directly from PC), you may hear the monitor signal twice, first one in time (directly from mixer?) and the second one late.

If it's not latency, then I suspect you do have an FX on somewhere. Your Alesis includes over 100 effects, have you double checked nothing is on?
MC does not have any hidden FX, if there's a reverb on somewhere, you should also see it.

How have you routed your projects? Does every track go "out" via the Master Bus, as is the recommended way?
How do you monitor - headphones or loudspeakers? Where do you get the monitoring signal - from computer or through Alesis?

Follow the signal through the whole system, either there's an FX or there's a feed back or somthing.

Correct me if I'm wrong: Alesis Multimix 6 is only a mixer, and does not include an audio interface?

The stock sound cards often have an effect on by default, usually a reverb-type effect.
Open the sound card's properties and see if that's the case.
Welcome to the forum.

Right click on the little speaker icon in the lower rght hand system tray (area where the date and time are. ) Poke around in there and see if you can find any effects checked.

MC 2 is pretty old (2002?) but you should be able to name your tracks and buses.
Double-click a track and you should be able to enter text, like 'Vocal 1' or 'Piano', a name that reflects whatever the track is.
Same for buses. If you have no stereo audio buses at the bottom, you can right-click in the bus pane and select Insert Stereo Bus, or go to the top, Insert>Stereo Bus. There may also be a little show/hide arrow control. Click it to show any buses that may already be there.
They are numbered by default, but you can name them just like the tracks. You may want all your drums in one bus, name that bus 'Drums', name another bus Vocals, differnt buses for each grouping of instruments.
Name one bus Master.
Now, click the outputs of the tracks and change them to the bus named for the type of track it is. Track X (Vocal) send to the Vocal bus, etc.
Now change all of the instrument/vocal bus outputs to Master.
That way you have tracks going to buses; buses going to Master; Master to sound card. Ideally, you should have no tracks going directly to the sound card, and no buses but the Master going to the sound card. This will give you greater control of all the summed volumes in one place, the Master bus.
Here's an image of a console view showing how I routed all the tracks and buses.
 

 
That won't fix your reverb problem, but we're thinking about it!