I can't answer the first question (whether this card is all you need or just a temporary one), but for the others:

Firstly, I assume you're talking about the BVG Umweltkarte (the other susbcriptions are only available to students/pensioners and have their own rules). It's certainly not true that the ticket is only valid for the person whose name is on the card - lending to friends and relatives is explicitly permitted:

Übertragbarkeit: Das Abo ist nicht personengebunden und kann an Freunde und Verwandte übertragen werden. Einfach fahrCard in die Hand drücken und los geht’s.

rough translation (gisted, not intended to be word-for-word):
Transferability: The subscription is not bound to a specific person and can be passed on to friends and relatives. Just give them the card and off they go.

and for your second question:

Kostenlose Mitnahme: Von 20 Uhr bis 3 Uhr sowie am Wochenende und an Feiertagen fahren eine Person jeden Alters und bis zu drei Kinder (bis 14 Jahre) kostenlos mit.

rough translation:
Companion travels free: From 8pm until 3am, and all day on weekends and holidays, one person of any age and up to three children can travel with you for free.

The card you would get in the mail is the same kind of plastic card that they gave you in the store. It should say fahrCard on it.

We can't know how long this card is valid though.

Your best option is to either call BVG, or take a friend who speaks German and go to their office.

Typically you would order the subscription, which starts from the 1st of the next month, then go to a store to get a temporary paper ticket called a Startkarte. This is a ticket you can buy for the remaining days until the start of your subscription, at the same price per day as you pay for during your subscription.

It might be that they have changed this from paper to a plastic card since I did it last in 2015.

You can find additional information on https://www.bvg.de/de/abo-online, which is only available in German.

You can also ask BVG on Twitter at https://twitter.com/BVG_Kampagne, where they will respond in English, and usually fairly quickly. Refer to your post here on Twitter and they should sort you out.

The other two answers are perfectly valid so far. I'd like to give some clarification on the "fahrCard" itself as well as some references.

The VBB

The VBB is the Verkerhsverbund Berlin Brandenburg, which is

the public transport authority covering the federal states of Berlin and Brandenburg – the capital area of Germany.

Source

The S-Bahn Berlin GmbH aswell as the BVG operate under their umbrella.

So, when you buy a ticket, monthly or yearly subscription, that ticket complies to the VBB tariffs and policies.

The VBB fahrCard

Benefits of the VBB-fahrCard

• Easy usability: Once on board, wave the card over the reader or present card to ticket controllers.
• Increased security: In the event of theft or loss, your chip card will be blocked immediately after you’ve made a report.
• Chip card replacement: In the event of loss, theft, or damage, you will receive a new VBB-fahrCard right away via the transportation company your subscription contract is with.
• More durable ticket: The plastic card is more durable than the previous paper card. The average life span of a chip card is about 4 years.
• Privacy protection: The VBB-fahrCard only stores data previously included on the paper ticket or customer card. Data on how often you travel will not be stored.

Source

Note: to my personal experience, the last cards lasted only about 2 years with light stress (bending) on them.

The card itself is valid for several years or until your subscription ends. My recent card was issued in October 2019 and on the card the printed validity period is "03/2024".

You can check your card data, validity and readability in some service centers of the Deutsche Bahn. One service center that I know is at Friedrichstraße next to the escalators near exit Friedrichstraße. It's a small terminal next to the ticket machines. Just touch the card onto the terminal.

The subscription recipe lists the data that is stored on your card and you can read the data on the card with any RFID/NFC reader app on a modern smartphone.

The data roughly includes:

• the card ID (related to your subscription)
• the tariff (VBB-Umwletkarte, for example)
• the tariff area (A/AB/ABC etc.)
• the card validity range

There's also a FAQ on their site: fahrCard FAQ

The fahrCard was introduced around 2015.

The VBB-Umweltkarte (yearly subscription)

• You can take 1 adult and 3 children with you after 8pm on weekdays and all day at weekends.
• This ticket is transferable, i.e. another person can use it.

Source

Unfortunately, I was not able to find these informations on the VBB nor on the BVG sites in English language.