Forget Mycelium, I will just import those keys into Electrum but even then how will I be able to run the Electrum Server ?
I tried to search it on google but the whole process seems to have so many steps.
Electrum private server is pretty much the most straightforward and least resource intensive way of going about it.
Can I like use my mobile phone wallet to push transactions to mempool having my full node on a different machine connected to my wallet on mobile phone.
The purpose of your full node is mostly as a data source. You can connect to any full node to push a transaction.
I've used this wallet (
https://play.google.com/store/apps/details?id=de.schildbach.wallet&hl=en) which allows you to connect to specific nodes only.