Is there any way to hide taskbar labels and not combine taskbar buttons?

"It's something of a hack job, but this might do the trick! It requires doing a bit of registry editing to shrink the taskbar icons so that the labels auto-hide.

Set taskbar options

Open Taskbar properties.
In the ""Taskbar Appearance"" group, change the ""Taskbar Button"" option to ""Never combine"" if you don't want stacking.

Adding/editing the registry entry

Press Windows Key + R to open the run command
Type int ""regedit"" (without quotes) and press OK
Go to HKEY_CURRENT_USER -> Control Panel -> Desktop -> WindowMetrics
Find the ""MinWidth"" entry.
If it is not there, right click on an empty space and select New -> String Value. Name this entry ""MinWidth"" (without quotes)
Double click on the MinWidth entry. If you want just the buttons to show, set this to 38 if you're using small buttons, 52 if you're using large buttons (or 54 in Windows 8). Otherwise any value above 38 will work. If you set a value below 38, the buttons will behave oddly. If your main display (on Windows 8.1 or later) has non-standard DPI (e.g. it's Retina or 4k display) you might need to multiply these numbers by the factor of 1.25, 1.5 or 2. Otherwise buttons will shrink / animate incorrectly.
Log off and log back in or restart to see the changes.

Deleting/Uninstalling

Press Windows Key + R to open the run command
Type int ""regedit"" (without quotes) and press OK
Go to HKEY_CURRENT_USER -> Control Panel -> Desktop -> WindowMetrics
Find the ""MinWidth"" entry and delete it.
Log off and log back in or restart to see the changes."

"This is an old question, however as Win 10 is still missing this option, I'm willing to leave this answer here:

Under taskbar settings, choose ""Always, hide labels""

Use one of the following options to disable taskbar combining:

option1: edit the registry

Go to the following Registry key:

HKEY_CURRENT_USER\Software\Microsoft\Windows\CurrentVersion\Policies\Explorer

On the right, create a new 32-Bit DWORD value NoTaskGrouping
Set its value data to 1 in decimals
Restart Windows 10

option2: edit group policy editor

Launch the Group Policy Editor app (gpedit.msc)
Set the option ""User Configuration \ Administrative Templates \ Start Menu and Taskbar \ Prevent grouping of taskbar items"" to Enabled

For more illustrations you could check this link: winaero.com/disable-taskbar-button-combining-windows-10/
Regards,
Bishoy"