Saving Logo with Correct Dimensions

Create a new document. Your settings should look like this:

Paste your logo onto the artboard, select it. Open the Transform panel.

Click the lower left point on the reference point icon. To avoid distorting your logo, in case your logo is not the exact dimensions of the artboard, click the chain icon on the right. Type in the Width and Height dimensions.

If the logo is not lining up with the lower left corner of the artboard, keep the logo selected and open the Align panel. In the lower right corner, select Align to Artboard.

Then click the Horizontal Align Left icon, and then the Vertical Align Bottom icon.

Choose File > Export. You can either choose Export for Screens or Save for Web (Legacy). In the screen captures below I am using Save for Web (Legacy).

In the upper right, choose PNG-8 or PNG-24, and check Transparency. (Do not choose JPG because it does not support transparency.) For Matte color, you can choose the color of the web page background. If you don't know the web page background color or if the logo will be used with backgrounds of different colors, choose None.

At the top of the screen, I like to choose 2-UP so that I can see the original on the left and the optimized version on the right. Below the preview you will see the file size.

If the file size exceeds the required 64MB, then select the drop down list in the upper right and choose Optimize to File Size.

Type in the desired file size and click OK.

Then you're ready to click the Save button.

By the way, a file size of 64MB is HUGE for the web I wonder if they meant 64K?

Create a new file 180w x 60h pixels.

Take your vector logo and proportionally scale it to fill the artboard either by width or height (whichever hits the edge of the artboard first). Next, center it within the artboard.

Export is as png. In the png export window, change the background color in the dropdown menu to "transparent".

Without seeing anything I would recommend that you start with the documentation so you get to know the tools:

How to scale, shear, and distort objects in Illustrator