Can you go through all 7 purifications of insight with walking meditation?

Yes, there have been some even mentioned in the Canon. But that does not mean that all are able in this way. Maybe useful:

The Benefits of Walking Meditation, by Sayadaw U Silananda (1996; 8pp./23KB) A concise outline of the practice of walking meditation as it is taught in the contemporary Burmese satipatthana "noting" tradition.

Ohh, as Atma has just seen, that Ven Bhante Yuttadhammo had given an answer, there are also good and helpful notes in his Book: Chapter Three: Walking Meditation

Can you go through all 7 purifications of insight with walking meditation. Can you also go through the 7 purifications of insight by focusing on your steps ...

Yes. Answer: https://buddhism.stackexchange.com/a/8282/295 already covers this aspect already.

... and saying to yourself "stepping, stepping"?

There are many view from different teachers and traditions on this and according to some this is not a valid way to do walking meditation

In walking meditation you have to look at either:

• look at changes in your posure, or
• sensation in your body when you walk

Look at Changes in Your Posure

When you are looking at your you use your body consciousness. If you look at the type of minds that arise from body consciousness they are always accompanied by 3 sensation which are either pleasant, unpleasant or neutral. So when you introspect the posture you are in your essentially knowing your posture thought these sensations. There is no mind state devoid of sensations.

Sensation in your Body when you Walk

When you do walk there are other sensation like the cloths touching your body. Wind brushing your legs. And other like heat or cold, perspiration running down your legs. You can keep being aware of the arising and passing of these sensations.¹

Why not to use Metal Recitation

Any labeling is constructed in a spoken language hence is verbal formation. E.g. Stepping. According to the Abhidhamma this is called nama pannatti₂. Any mental construction can lead corruptions of insight as they are can be tainted by any of the Vipallasa³. Also say you were Walking and stopped then you are standing then the meaning of walking is lost, hence language constructs not realities⁴.

If you see this process as cognising the words "walking", "stepping" then this can become insight meditation again. Here you can to see the arising and passing of sensation associated with cognising.

But having said this there are some Schools of Buddhism which does use techniques like this.

¹

Iriyāpathapabbaṃ—Postures of the Body

Iriyāpatha are postures of the body.

gacchanto vā ‘gacchāmī’ti pajānāti, ṭhito vā ‘ṭhitomhī’ ti pajānāti, nisinno vā ‘nisinnomhī’ti pajānāti, sayāno vā ‘sayānomhī’ti pajānāti.

When walking (gacchanto), a meditator knows well ‘I am walking’ (‘gacchāmi’). Similarly, whether standing (ṭhito), sitting (nisinno), or lying down (sayāno) a meditator knows this well. This is just the beginning. In the sentence that follows, not "I", but just "body" is known well in whatever posture (yathā yathā paṇihito).

Yathā yathā vā panassa kāyo paṇihito hoti, tathā tathā naṃ pajānāti.

Then, in a repetition of the same sentences, the body is observed inside, outside, and both inside and outside simultaneously. Arising is observed, then passing, then both together. Actually it is the sensations that are observed as arising and passing away, because sampajāñña, the understanding of anicca, has to be present, as in every chapter. Awareness follows that ‘This is body,’ and that it is not "I." This is established with wisdom. Then mere understanding and awareness follow, without any base to hold. There is nothing to grasp.

Iti ajjhattaṃ vā kāye kāyānupassī viharati… ‘atthi kayo’ ti…na ca kiñci loke upādiyati.

Source: Discourses on Satipatthana Sutta by S.N.Goenka

₂ The Theravada Abhidhamma: Its Inquiry into the Nature of Conditioned Reality by Y. Karunadasa, Chapter 2

³ See Vipallasa Sutta for definition.

⁴ SN 22.62 Niruttipatha Sutta

Yes, of course.

After reciting the Pātimokkha, it seems, on the Uposatha day of the full moon, one of the two elders who were brothers went to his own dwelling place surrounded by the Community of Bhikkhus.

As he stood on the walk looking at the moonlight, he calculated his own vital formations and he said to the Community of Bhikkhus, “In what way have you seen bhikkhus attaining Nibbāna up till now?”

Some answered, “Till now we have seen them attain Nibbāna sitting in their seats.” Others answered, “We have seen them sitting cross-legged in the air.”

The elder said, “I shall now show you one attaining Nibbāna while walking.” He then drew a line on the walk, saying, “I shall go from this end of the walk to the other end and return; when I reach this line I shall attain Nibbāna.”

So saying, he stepped on to the walk and went to the far end. On his return he attained Nibbāna in the same moment in which he stepped on the line.

-- Vism. VIII.244 (Nyanamoli, trans)

Note, that the seven purifications are not all related to insight directly. They are simply called the seven purifications, not the seven purifications of insight.