Is it safe to sweep all three Bitcoins into Coinomi, using my private key?
That depends... Is your phone clean? Do you take safety precautions like a good password, a safe backup, and making sure nobody can see your private keys and seed while doing so?
It also depends on how much value you're storing on your phone. A year ago, I'd totally trust my phone with 0.1
BTC, as it was worth less than $100. Now that's $1500, which is much more than I trust my phone with (and much more than I need for daily use).
Basically, I only trust my phone with amounts I'm willing to lose.
I thought Coinomi was an online wallet. Is Coinomi the cheapest one regarding fees?
Fees depend on how much competing transactions are willing to pay for a confirmation. But Coinomi allows you to set your own fee.
I've also heard about Mycelium. Do you know which one of all the wallets that has the lowest fee for doing this?
Mycelium also allows you to set your own fee. In both wallets you can set it low enough to never confirm, so it's entirely up to you how much to pay.
And is it possible to do while I am offline?
No.