I noticed that you asked many questions about what Dvaita teaches. You should know that Dvaita is just one specific system of the philosophy of Vedanta promulgated by the Madhva (Madhvacarya or Anandatirtha) who was the leader of one Vaishnava tradition (sampradaya) in the 13th century. See the Wikipedia article on Madhvacharya.
However if you are interested in Vaishnava systems of Vedanta in general you should ask your questions like "According to Vaishnava philosophy ..."
Is Vishnu the only real God?
All the gods are real, but among them there is one supreme god, it's Lord Krishna.
There are no unreal gods!
If yes, then why did he pray as Krishna to another demi god when he himself had warned against it?
He prayed to gods such as Shiva to show an example that you can pray like that, but in the Gita He explained to Arjuna that only less intelligent people think that they should worship gods in heaven and not paramatma, Lord Krishna, instead.
Bhagavad gita 7.23 — Men of small intelligence worship the demigods, and their
fruits are limited and temporary. Those who worship the demigods go to
the planets of the demigods, but My devotees ultimately reach My
And He also said in the verse 9.23 that the devotees of other gods perform worship "in a wrong way", as you have quoted. Some people are not capable to worship paramatma, Lord Krishna, and they are not eligible to get liberation (moksha), and Lord Krishna showed an example to those people to worship some other gods. It is better to worship any god than not worship at all. That is why Lord Vishnu or Krishna sometimes worship some gods, just to show an example to people who are not capable to worship the Supreme God, Lord Krishna.
Would Dvaitins recommend praying to Shiva?
No, they would not recommend worship of anyone except paramatma, the Supreme Absolute, Brahman, Lord Krishna. In the Bhagavad gita Lord Krishna recommends worship of Lord Krishna only as the best and ultimate type of worship.
Their Parmatma (Vishnu) prays to another Parmatma (Shiva)...There is not one Parmatma but many.
That what you said is contrary to the teaching of Vedanta. The Upanishads say that there is only one paramatma in all the beings:
eko devaḥ sarva-bhūteṣu gūḍhaḥ sarva-vyāpī sarva-bhūtāntarātmā
karmādhyakṣaḥ sarva-bhūtādhivāsaḥ sākṣī cetā kevalo nirguṇaś ca
"He is the one God, hidden in all beings, all-pervading,
the self within all beings, watching over all works, dwelling in all beings, the witness, the perceiver, the only one, free from qualities." (Svetasvatara Upanishad 6.11)
If you read carefully this verse you should notice that it says there is only "one God" (eko devaḥ) and also "the only one", who is the paramatma hidden in all beings (sarva-bhūteṣu -- in all living entities; gūḍhaḥ -- hidden), and also that He is the self (paramatma) within all beings (sarva-bhūta -- all living entities or jivatmas; antah-atma -- the Supersoul, the Self, Supreme Soul, paramatma). So, there are not many paramatmas, but there is only one paramatma.