will Airdrop run on High sierra V 10.13.6 MacBook Pro (late2010)

Hello Titus...

To another Mac yes, but not to an iDevice...

AirDrop system requirements

To share between a Mac and an iPhone, iPad, or iPod touch, you need any of these devices and operating systems:

Mac from 2012 or later (excluding the mid-2012 Mac Pro) using OS X Yosemite or later

iPhone, iPad, or iPod touch using iOS 7 or later

To share between Mac computers, both computers must be one of these models:

MacBook Pro introduced in late 2008 or later, excluding the MacBook Pro (17-inch, Late 2008)

MacBook Air introduced in late 2010 or later

MacBook introduced in late 2008 or later, excluding the white MacBook (Late 2008)

iMac introduced in early 2009 or later

Mac mini introduced in mid 2010 or later

Mac Pro introduced in early 2009 (model with AirPort Extreme card) or mid 2010

iMac Pro (all models)

https://support.apple.com/en-us/HT203106

It appears that in addition to those requirements, Catalina has dropped support for legacy AirDrop. So, you wouldn't be able to drop to a Catalina Mac.