BuddyPress.org

I’m not aware of any that integrate into BuddyPress.

Well, there are several Group chat rooms integrated well with BuddyPress.

When you choose, you need to consider the following:
1 – You can install one on your server, but if you have more than 10 chatters, that will probably load your server. Most shared hosting services, do not allow to install a chat on the server as it is high cpu/mem consumer. In addition, sharing files is dead slow.

2 – I recommend a group chat hosted service, it will offload all activity to a remote server, bandwidth is not a consideration. They work with CDN, so everything is pretty fast.

As for your question:
I am very much in favor of RumbleTalk chat plugin, works very well and integrate with your users base.

– Ability to send any files (anything from videos, photos, music, etc.)
Yes -You can upload files, pdf,excel..etc, I think any file, but they have size limit
– Ability to text chat
Yes
– Ability to video or voice call to a friend
Yes – they have private chat (you need to allow it in the settings), you must be in ssl to use it (WebRTC)
– (Optional) ability to add other friends or users to the chat
Yes – you can allow facebook/twitter friends
– (Optional) ability to screen share (desktop sharing)
No – no screen sharing

It is not that pricey. But what I see that you need to install it on your dedicated server, so add it to your calculation.

using this in my bp setup https://wordpress.org/plugins/wise-chat/ and works very well

if you want p2p individual chat like FB then you need to buy the pro version which is $49

Mirrorfly , a white-label chat solution to build instant messaging app for any business with features & customizations that runs on Android, iOS, & Web.

Mirrorfly SaaP Solution – One Time Payment & customized Solutions
Messaging & Collaboration – Connect, Chat & Collaborate Beyond Platforms

Private 1-to-1 Chat
Push Notifications
Message Broadcast Option
Cross-platform Messaging

Voice Communication – Make Real-time Conversations Anytime, Anywhere
1-to-1 Call
Conference Calling
Voice Call Recording
Cross-platform Calling
Cross-platform Calling
Interactive Voice Response (IVR)

Video Communication -Explore HD Video Experience You Demand
One-to-one
Multi-party
Video Call Recording
Screen Sharing
Picture-In-Picture
Whiteboard

Explore more features here

BP Better Messages – is a fully featured replacement for standard BuddyPress Messages and also can work as private messaging system for WordPress when using without BuddyPress.
Plugin is fully backward compatible with BuddyPress Messages.

BP Better Messages