In short: Yes to both.
What follows is a very simplified explanation. More details linked below.
When CAs sign certificates, they do not only sign the public key of a website e.g., but actually a whole lot of metadata. This meta data e.g. includes when the certificate expires or so.
In our case, this is saved in a data format defined as X.509.
It also includes certain "constrains", i.e. limits on what the cert can do. The Wikipedia article lists an overview, but what is important for our case is: It also says whether that cert can sign other "(sub)certs" and thus "certify" them or no.
Both the root CA and intermediate CAs/certs have this set.
Thus, they are allowed to sign other certs.
A chain of trust
What browsers then do is take the cert delivered by the website (for
https://example.com e.g.) and try to build a so-called certificate chain, i.e. "chain" that cert up until it reaches the root cert.
You already draw it, but you missed that last cert I've just talked of, which is usually called the leaf certificate. (Likely, because of the tree data structure
|---LEAD_CERT valid for example.com (isCA=no)
Obviously, the leaf cert must not have the permission to sign other certs.
In practice this allows many many things you would not think of when we would only have one CA and then the leaf cert that is directly signed by then.
E.g. CAs sometimes re-sell their selling infrastructure and allow other companies to sign their own certs with an intermediate certs. (but there are still technical and legal things they have to adhere to, e.g. AFAIK they just offer an API)
Or, as the private keys for root certs are e.g. usually kept in HSMs, for productive use they use an intermediate CA, so they could switch that out if they really gets in trouble (their intermediate CA is compromised).
Or you could even do crazy things such as cross-signing, where one CA signs the root for another CA, so it gets faster acceptance in browser. Let's Encrypt e.g. did this – and also has an interesting and well-documented structure of their intermediate certs, if you want to see a "real world" example.
If you want to read more, I really recommend you to look into: