Check rust code with dprint / clippy

This issue collects infos from multiple places and some research I have done.
# What
At the moment dprint is already used in the CI to check various file format for consistent formatting ([config](https://github.com/google/comprehensive-rust/blob/main/dprint.json))
https://github.com/google/comprehensive-rust/blob/a8460036650ecb8f980a3081a9f09cfcc6466580/.github/workflows/build.yml#L30-L31

Pull requests https://github.com/google/comprehensive-rust/pull/2025 and https://github.com/google/comprehensive-rust/pull/2410 argue already for a check with cargo clippy to have an in-depth check of rust as well.

https://github.com/google/comprehensive-rust/pull/2025#issuecomment-2427392137 mentions that [clippy-driver](https://github.com/rust-lang/rust-clippy?tab=readme-ov-file#using-clippy-driver) could do the checking as in contrast cargo clippy as mentioned in https://github.com/google/comprehensive-rust/pull/2025#discussion_r1806091321

# clippy-driver
An easy implementation would include clippy-driver as an dprint-plugin-exec command like
```json
{
"command": "clippy-driver --edition 2021 -Cpanic=abort {{file_path}}",
"exts": ["rs"]
}
```
As already mentioned in https://github.com/google/comprehensive-rust/pull/2413#issuecomment-2421834318 this creates some problems as dprint only works on individual files while cargo clippy looks at the entire context (e.g. dependency crates).

https://github.com/google/comprehensive-rust/pull/2413#issuecomment-2427377949 argues that the code examples inside markdown files are pretty self-contained as they rely on playground and its crates.

## Problems

And while that might be true, it is not as easy as it looks, because the rust playground provides the top 100 most downloaded crates from crates.io (https://play.rust-lang.org/help#features-crates) ([explicit list](https://github.com/rust-lang/rust-playground/blob/main/compiler/base/Cargo.toml)). These dependencies are not considered by clippy-driver

### dependency crates
An example that is using that can be found at [concurrency/async-exercises/chat-async/src/bin/client.rs](https://github.com/google/comprehensive-rust/blob/50828d15d8a3cc98259f4e6350030c7464032042/src/concurrency/async-exercises/chat-async/src/bin/client.rs).

Output from `clippy-driver --edition=2021 src/concurrency/async-exercises/chat-async/src/bin/client.rs`

```
$ clippy-driver --edition=2021 src/concurrency/async-exercises/chat-async/src/bin/client.rs
error[E0433]: failed to resolve: use of undeclared crate or module `futures_util`
--> src/concurrency/async-exercises/chat-async/src/bin/client.rs:17:5
|
17 | use futures_util::stream::StreamExt;
| ^^^^^^^^^^^^ use of undeclared crate or module `futures_util`

error[E0432]: unresolved import `futures_util`
--> src/concurrency/async-exercises/chat-async/src/bin/client.rs:18:5
|
18 | use futures_util::SinkExt;
| ^^^^^^^^^^^^ use of undeclared crate or module `futures_util`

error[E0433]: failed to resolve: use of undeclared crate or module `tokio`
--> src/concurrency/async-exercises/chat-async/src/bin/client.rs:20:5
|
20 | use tokio::io::{AsyncBufReadExt, BufReader};
| ^^^^^ use of undeclared crate or module `tokio`

error[E0432]: unresolved import `http`
--> src/concurrency/async-exercises/chat-async/src/bin/client.rs:19:5
|
19 | use http::Uri;
| ^^^^ use of undeclared crate or module `http`

error[E0432]: unresolved import `tokio_websockets`
--> src/concurrency/async-exercises/chat-async/src/bin/client.rs:21:5
|
21 | use tokio_websockets::{ClientBuilder, Message};
| ^^^^^^^^^^^^^^^^ use of undeclared crate or module `tokio_websockets`

error[E0433]: failed to resolve: use of undeclared crate or module `tokio`
--> src/concurrency/async-exercises/chat-async/src/bin/client.rs:23:3
|
23 | #[tokio::main]
| ^^^^^ use of undeclared crate or module `tokio`

error[E0433]: failed to resolve: use of undeclared crate or module `tokio`
--> src/concurrency/async-exercises/chat-async/src/bin/client.rs:36:9
|
36 | tokio::select! {
| ^^^^^ use of undeclared crate or module `tokio`

error[E0433]: failed to resolve: use of undeclared crate or module `tokio`
--> src/concurrency/async-exercises/chat-async/src/bin/client.rs:30:17
|
30 | let stdin = tokio::io::stdin();
| ^^^^^ use of undeclared crate or module `tokio`
|
help: consider importing this module
|
17 + use std::io;
|
help: if you import `io`, refer to it directly
|
30 - let stdin = tokio::io::stdin();
30 + let stdin = io::stdin();
|

error[E0752]: `main` function is not allowed to be `async`
--> src/concurrency/async-exercises/chat-async/src/bin/client.rs:24:1
|
24 | async fn main() -> Result<(), tokio_websockets::Error> {
| ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ `main` function is not allowed to be `async`

error[E0433]: failed to resolve: use of undeclared crate or module `tokio_websockets`
--> src/concurrency/async-exercises/chat-async/src/bin/client.rs:24:31
|
24 | async fn main() -> Result<(), tokio_websockets::Error> {
| ^^^^^^^^^^^^^^^^ use of undeclared crate or module `tokio_websockets`

error: aborting due to 10 previous errors

Some errors have detailed explanations: E0432, E0433, E0752.
For more information about an error, try `rustc --explain E0432`.
```

### complex code
A solution should also take into account e.g. [mdbook-course](https://github.com/google/comprehensive-rust/blob/50828d15d8a3cc98259f4e6350030c7464032042/mdbook-course) and other directories that are not as simple.

## Other things

I also noticed that clippy-driver leaves some trash behind, etc. a binary `dining-philosophers", when`dprint` is checking that example it also gives an error that is not explaining a lot:

```
dprint check "src/concurrency/sync-exercises/dining-philosophers.rs"
Error formatting /home/kerscher/workspace/comprehensive-rust/src/concurrency/sync-exercises/dining-philosophers.rs. Message: The original file text was greater than 100 characters, but the formatted text was empty. Perhaps dprint-plugin-exec has been misconfigured?
Had 1 error formatting.
```

Additionally it feels like abusing dprint a bit here. dprint is a *formatting* tool while clippy is a code *linter*. What is dprint fmt supposed to be doing?

# cargo clippy
the workspace [Cargo.toml ](https://github.com/google/comprehensive-rust/blob/a8460036650ecb8f980a3081a9f09cfcc6466580/Cargo.toml) already defines all members of the workspace that are relevant.
We could use the [workspace.lints-table](https://doc.rust-lang.org/cargo/reference/workspaces.html#the-lints-table) to define what clippy rules we want and a cargo clippy run at the workspace level would check all the projects in that workspace