You would have to put in a 3rd party solution such as Google Analytics to get your results as I concur with Blake.

TL;DR: it's possible with a customization outside the normal toolset.

The buttons you would like to be monitored all appear to be on the Thank You Page.

This page is governed by the MVC framework and is in the View titled Submit found at: {install directory}\Laserfiche\Laserfiche Forms\Forms\Views\Form\Submit.cshtml.

Here you will find all the buttons you mention in denoted in razor syntax.

	<div id="submission-toolbar" class="hidden btn-toolbar text-center" role="toolbar">
		<div class="form-group">
			@if (Model.StartEvent.AllowDownloadSubmission == true)
			{
			<div class="btn-group btn-group submission-summary-btn">
				<button id="sub-download-btn" class="btn btn-primary" onclick="downloadSubmission()">@Resources.STR_BTN_DOWNLOAD</button>
			</div>
			}
			<div class="btn-group btn-group submission-summary-btn">
				<button id="sub-print-btn" class="btn btn-default" onclick="window.frames['submission_iframe'].focus(); window.frames['submission_iframe'].print()">@Resources.STR_PRINT</button>
			</div>
			@if (Model.StartEvent.AllowEmailSubmission == true)
			{
			<div class="btn-group btn-group submission-summary-btn">
				<button id="sub-email-btn" type="button" class="btn btn-default">@Resources.STR_EMAIL</button>
			</div>
			}
		</div>
	</div>

You can then, with diligence & discretion, plug in javascript to have it monitored by Google Analytics.

This is globally editing every thank you page generated by LF Forms and if not careful could break your applications. To apply tracking per forms business process, you'll have to tie in the business process ID.

*We're venturing off the beaten path and far outside the normal sandbox Laserfiche has worked hard to support for us.  It's up to you to maintain and support it along with the possibility it may be overwritten when applying hotfixes and upgrades. Keep your own backups and documentation of anything edited here. Do not prevent the trademark and logo from displaying.

I'm making an assumption this is allowed due to their views on editing WebLink pages.

Cheers,

Carl