Can I use the phone number assigned to my iPad Air to send messages and make phone calls?

No. In more detail...

An iPad, whether a Cellular model or not, is not capable of directly accessing Cellular Voice or SMS/MMS Messaging features. An iPad with either WiFi or Cellular connectivity alone can only access IP-data services. If you have added an eSIM or physical DataSIM and Cellular plan to a WiFi+Cellular model of iPad, the number associated with the eSIM/SIM card is only used to identify your account for billing purposes. This number is not used for making calls or sending messages. An iPad is not, and can never be, a standalone substitute for an iPhone.

However, if coupled with an iPhone, the capabilities of iPad significantly broaden to include access to Cellular Voice and Messaging services of the associated iPhone - relayed to the iPad over WiFi using Apple’s Continuity features. For the most part, an iPad using Continuity is able to access Cellular calling features, of the iPhone, when both the iPad an iPhone are in close physical proximity and are connected to the same WiFi network - but this is not the end of the story.

Now, if your both your Cellular Carrier and your calling plan support WiFi Calling, your iPad can make and receive calls independently of the iPhone - even when the iPhone is switched off, elsewhere, or using a different WiFi network. Not all carriers support full WiFi calling.

The settings that you need to change/verify on your iPhone and iPad are:

On your iPhone

Settings > Mobile Data >WiFi Calling > 

• WiFi Calling on This iPhone - set to ON
• Add WiFi Calling for Other Devices - set to ON

Settings > Mobile Data > Calls on Other Devices > 

• Allow Calls on Other Devices - set to ON
• [Allow Calls On] - explicitly enable your new iPad from the list of devices.

On your iPad

Settings > FaceTime > Calls from iPhone > Calls from iPhone - set to ON

Additional information about Continuity and the relevant Cellular services can be found here:

Continuity

Work across all of your devices seamlessly – Apple Support

SMS/MMS Messaging

Forward SMS/MMS text messages from your iPhone to your Mac or iPad - Apple Support

WiFi Calling

Make a call with Wi-Fi Calling – Apple Support

Making and receiving calls on iPad

Make and receive phone calls on iPad - Apple Support

When correctly configured, the iPad FaceTime App provides access to Cellular Voice Calling features - whilst Messages supports SMS/MMS messaging services.