I am wondering if there are examples of full application stack based on Kubernetes

Althought you do have some examples (random example and git of part of random example) you most probably won't get silver bullet there since a job of bringing up and configuring full application stack is often dependent on details of your exact cluster and your use case (PVC for example will have different manifests if you run on bare metal, aws or gke... Security policies might differ etc) so most probably you will have to do a lot of small tweaking yourself and experiment a bit on your own to get it exactly right for your cluster and use case. Finally, you might want to check for helm charts as well.

And is it a good idea to have services like PostgreSQL and Solr on Kubernetes?

Depends :) If your database load is brutal then those 11 miliseconds of overhead that k8s is introducing might be a show stopper. If you are not squeezing every last bit of performance out of database/indexing engine/no-sql store then you are good to go with k8s. We run multiple mysql and postgresql database in production in k8s (You have to be careful about PV I/O capabilities though). So it might or might not work for you.