php mail

It should work in Chrome. Have you emptied the chrome browsers cache. Bens post shows it working in the Chrome browser.

You should not have any issues with reducing the top/bottom padding on the 'alert_message' css from 12px to 4px:

Have you uploaded the revised css file or maybe its a browser cache issue, where you are seeing a previous version?

.alert-message {
color: black;
padding: 4px 12px;
border-radius: 4px;
text-align: center;
margin: 10px 0px 20px 0px;
background-color: #7ec352;
font-family: 'Neuton', serif;
font-size: 15px;
}

Similarly as Ben has pointed out there should be no issues with adding some margin to the 'btn-green' css:

.btn-green {
color: #fff;
padding: 4px 7px;
font-family: 'PT Serif Caption', serif;
font-size: 13px;
font-weight: 400;
letter-spacing: 0.5px;
border-radius: 4px;
background-color: #7ec352;
border: 0.95px solid black;
display: block;
margin: 20px auto;
}

Q1

Q2. I do not see the alert message to give you an answer.

Off Topic:

If you are using the Chrome browser, right-click th page and choose Inspect

This will give you the Developer's window as in above.