Sentinel-2 L2A imagery

Is it necessary to get a plan or just a free acount?

Using EO Browser you can search for the scenes and then get links to the original products either stored on AWS or in Copernicus Open Hub. Up to this part, you do not need any subscription, not even registration as long as there is no commercial use. If you are using EO Browser account for commercial use, you need either Basic or Enterprise account.
The download from these two systems is not managed by us, you need to arrange it separately. But you might make use of the links.

If you want to use Sentinel Hub API, you need a paid subscription…

Can I get full scenes?

Using the links, as described above, you can get access to original scenes, so full.
By using Sentinel Hub API you are limited to 2500x2500 px.

I have used Sentinel Hub before to get Planet Scope imagery. Is it the same procedure to download Sentinel-2 images? First they have to be ordered in EO Browser and then downloaded or is there a better way to do it with Sentinel-2?

There is no need to order data in case of Sentinel-2. You can do it directly as shown in these examples:

docs.sentinel-hub.com

Examples for S2L2A

Use these CURL and Postman examples to access atmospherically corrected Sentinel-2 L2A data with Sentinel Hub API.

In case I subscribe to Sentinel Hub, are there different plans for Sentinel-2 and Planet Scope or I can access both of them? (I already have a Planet Scope subscription)

If you already have a Planet Scope subscription, you only need Sentinel Hub subscription and then you can download both Sentinel and PlanetScope data. There is no additional charge for this part. You will need to inform us about it, so that we set-up your account.