Are there any similar characteristics between Nama and Rupa?

OP: We can define location, size, velocity for rupa. Can we define these for Nama ?

No.

OP: "after the death, mind leaves the body and step into a new body" ?

This arises and passes. When one state passes away the next state arises. At death when the last though moment passes the next the rebirth linking though moment arises in another moment. Distance is immaterial.

The answer to your question is yes, there are three characteristics they have in common they are illness, are impremanent and are not self. There are more; they should not occur, they should not be, and so on with inferable from the first 3; they do not belong to a self, are empty of a self, are unpleasant, so forth and so on inferring from the first 3.

You seem to assume that the word 'body' as is used in conventional language & terms in the world has the same meaning the word Rupa as is used throughout the Sutta across the contexts, in particular the context of Dependent Orgigination. This isn't the case and a quick glance at the Vibhanga section on Form will make this obvious.

I suspect you have come to hold an assumption that namarupa means mind & body, wherein body is as is spoken of in conventional terms of newtonian physics.

Buddha spoke of that body you seem to speak of, which rots away, flesh & bones, in conventional terms, also referred to in pali as 'Kaya' but he also spoke of 'Rupa' in the context of 'namarupa' in Dependent Origination, that word is hard to see the meaning of.

OP: We can define location, size, velocity for rupa. Can we define these for Nama ?

Location, size, velocity, shape, time, ... all these are conventional. According to the ultimate truth neither rūpa nor nāma has those characteristics.

OP: Can we say "a moving mind", "mind is located inside the body", "after the death, mind leaves the body and step into a new body" ?

No, neither rūpa nor nāma can move. They arise, exist, and cease then and there. New rūpa may arise at the same place or nearby. They will also exist, and cease then and there. This is called "Deshāntarotpatti".

OP: Can we imagine "How mind looks like" ?

No one can see mind with eye consciousness. So you can't imagine "How mind looks like". But you can understand the behaviour and characteristics of your mind.

OP: Are there any similar characteristics between Nama and Rupa?

To answer your original question above, yes, there are similar characteristics between rūpa and nāma. Both rūpa and nāma are conditioned. All the conditioned things share three common characteristics. They are anicca (impermanence), dukkha (suffering), and anatta (not-self).

The best thing to remember the fire simile. The campfire become a grass fire. the grass fire becomes a bush fire. The bushfire becomes the housefire. Here the fire is similar to the consciousness (Nama). What fire depends on (grass etc) are the Rupa. When we take the fire with say grass it is called Namarupa.

We can define location, size, velocity for rupa. Can we define these for Nama ?

No, you can't do that in the same way. However, they are all in the difference between Nama such as wholesome, unwholesome, man's mind, girl's mind, etc.

Can we say "a moving mind", "mind is located inside the body", "after the death, mind leaves the body and step into a new body" ?

Both Nāma and Rūpa completely vanish where they arise immediately. Nothing move. However, there are many vanished origins can cause new Nāma Rūpa in the future.

Can we imagine "How mind looks like" ?

Yes, you can imagine of mind. However, no one can see mind by their eyes because mind has no color.

So, if you want to imagine about mind, you need to learn Abhidhamma and meditate Jhāna. If you attain Jhana, you can understand your mind easily. And you can learn everything easily because of Jhana. Jhana can develop your mind faster and brighter more than whenever you have been before 😊 And by the trick in Tipitaka, especially Abhidhamma, Nāma and Rūpa will clearly appear for you.