You can look into MetalLB, specifically into the Layer2 & Local traffic policy
(https://metallb.universe.tf/usage/)
You cannot assign IPs to the Pods but when you'll create a service type LoadBalancer (for example a http routing service like Traefik), MetalLB will help bind that service to the IP of the node.
As an example, you can see the external IP of the service Trafik is reported as my node's address - 192.168.1.201
NAME STATUS ROLES AGE VERSION INTERNAL-IP EXTERNAL-IP OS-IMAGE KERNEL-VERSION CONTAINER-RUNTIME
node02 Ready <none> 8d v1.20.2+k3s1 192.168.1.202 <none> Alpine Linux v3.13 5.10.10-0-virt containerd://1.4.3-k3s1
node01 Ready control-plane,master 8d v1.20.2+k3s1 192.168.1.201 <none> Alpine Linux v3.13 5.10.10-0-virt containerd://1.4.3-k3s1
For q2:
Of course you can, k8s doesn't take over the node. You ssh into it and run whatever you like.
For q1:
No.
NAMESPACE NAME TYPE CLUSTER-IP EXTERNAL-IP PORT(S)
default service/kubernetes ClusterIP 10.43.0.1 <none> 443/TCP
kube-system service/kube-dns ClusterIP 10.43.0.10 <none> 53/UDP,53/TCP,9153/TCP
kube-system service/metrics-server ClusterIP 10.43.254.20 <none> 443/TCP
kube-system service/traefik LoadBalancer 10.43.130.1 192.168.1.201 80:31666/TCP,443:31194/TCP,8080:31199/TCP
default service/whoami ClusterIP 10.43.61.10 <none> 80/TCP