MSSQL Transactional Replication
I wen through your post and here is my response (one-by-one) on the basis of what I got.

It's not really a problem if these processes run slow or get delayed

That's good if it suits you requirement.

But on the other hand this data should be available via a website and then the queries on the data should be quick

and

I was thinking about replicating parts of the processed data to a second server and running the website on that database.

In transactoinal replication SQL server allow to create index on the tables in subscriber database. So you want to use subscriber to generate reports, you can optimize you query and create indexes (on subscriber DB) that boosts your query performance to get quick response.

But if you want to make transaction at both side INSERT,UPDATE, DELETE then you should go for MERGE Replication.

I didn't understood rest part of you post. It would be good to write proper suggestion if you clarify it.

Thanks!