Does mnemonic phrase contain everything in my wallet including future keys? I am confused about something important ...
Most wallets allow wallet recovery using &quot;Mnemonic phrase&quot;, but (in general) with this phrase, do I get access to all my data ? E.g future generated addresses that will hold my bitcoins ?
Another way to ask: is this the only thing I need to store or do I also need to export my actual wallet/keys when I generate a new address or receive new coins ?
I know each wallet works differently but I think this would be a general question.
If it matters, I noticed the Mnemonic feature on Electrum and Coinomi wallets.
Edit: how can Coinomi give me access to different coins using same mnemonic phrase ? Does this mean that it is more like a password to my data (data stored by them) and if their wallet service becomes unavailable I don't really have access to any of my different coins ?
Generally, yes, mnemonic phrase contains enough data for software wallets to derive the keys. With the caveat.
Some software wallets imports are supporting only 12 word or 24 word combinations. Some support different BIPs and thus path derivation schemes (32/44/84) by default, and some only single one of them. Some have additional feature for a specific seed passphrase. All of the differences you can check on https://walletsrecovery.org/
And there are at least three popular mnemonic schemes:

bip39 - used in most cases
aezeed for lnd
electrum format

So to answer your question in practice - you have to install the wallet that uses the same seed format and path derivation as the one you've originally created for it to work.
Answering edit section:
Because BIP44 (which coinomi uses) defines derivation as:
m / purpose' / coin_type' / account' / change / address_index

You can derive multiple types of independent cryptocurrency paths from a single seed by changing coin_type parameter, creating a subtree for every coin type. This is a list of registered coins:
https://github.com/satoshilabs/slips/blob/master/slip-0044.md