We had a rather moody 10 year old dog (OD) in situ when we considered having a greyhound. We took OD to meet the prospective greyhound at the rescue centre and there was an introduction of sorts. OD was unfriendly as he was with all dogs but the greyhound was just shakey and didn't know how to react to a non-greyhound dog.
Reassured by the the rescue staff we took the greyhound home and then took them both for a short walk. The greyhound walked beautifully on the lead but was very nervous and our OD didn't do anything to make her feel welcome.
However within hours of them being together in the house our OD settled. There was no challenging for the top position from the greyhound - she just followed OD around as she hadn't got a clue what to do in a house. And that's how it continued for a further 5 years until OD died. Our greyhound adored OD and our OD seemed to thrive on being in charge.
Our greyhound lived a further 5 years and after a brief period of seeming lost she adapted to being our only pet.
A few things to mention:
On her second day our greyhound was rolling around happily on her back (as greyhounds do) when she got her paw stuck in the base of a radiator and was trapped and squealing. It took me a while to free her, and I must have hurt her in trying to do so, but there was no snapping or growling and her paw seemed uninjured. I was amazed that she trusted me.
Soon after our greyhound moved in I went out briefly and left both dogs in the kitchen. When I returned the greyhound had clawed the kitchen door badly clearly frightened of being confined. I never did that to them again.
Greyhounds are special; different from other dogs and splendid companions. They are not clever dogs and they require only minimal exercise. DS took ours jogging once. Greyhounds seemingly can't jog but they know how to relax and be as comfortable as possible. All they want is to feel safe and comfortable.
Consider what is important to you. I wanted one that would walk well on the lead - most do, but some don't. Most can't be let off the lead. I didn't want a reactive dog but was fine with a nervous dog. These dogs have never lived in a home or been loved. It takes time and understanding and is absolutely worth it. When our circumstances change we will have another greyhound, probably a girl again as they are smaller.