A menstruating woman entered ihram for ‘umrah and did sa‘i, then she did tawaaf after she purified herself

You did the right thing by entering ihram at the miqaat
when you were menstruating. The evidence that it is correct to enter
ihram when menstruating or bleeding following childbirth is the report
which says that Asma’ bint ‘Umays (may Allah be pleased with her) gave
birth when the Prophet (blessings and peace of Allah be upon him) had
halted at Dhu’l-Hulayfah (the miqaat for the people of Madinah) and was
intending to perform Hajj. She sent word to the Prophet (lettings and
peace of Allah be upon him) asking: What should I do? He said:
“Take a bath (ghusl), wrap your
private part in a cloth,
and enter ihram.” Narrated by Muslim (1218). 

You also did right by refraining from tawaaf when you were
menstruating. The Prophet (blessings and peace of Allah be upon him) said to
‘Aa’ishah (may Allah be pleased with her), when she got her menses during
the ‘umrah of her Hajj – as she was doing Hajj tamattu‘ i.e., ‘umrah
followed by Hajj, exiting ihram in between –: “Do what the pilgrims do, but
do not circumambulate the House until you are purified.” Narrated by al-Bukhaari
(1650) and Muslim (1211).

But you made a mistake by doing sa‘i and cutting your hair
before doing tawaaf, because the permissibility of doing sa‘i before tawaaf
applies only to Hajj, not ‘umrah, according to the more correct opinion.
Therefore ‘Aa’ishah (may Allah be pleased with her) did not do the sa‘i of
her ‘umrah when she got her menses. Exiting ihram and cutting the hair only
come after doing both tawaaf and sa‘i; before that it is not allowed, and a
compensatory sacrifice must be offered for that. 

Shaykh Muhammad ibn Saalih ibn ‘Uthaymeen (may Allah have
mercy on him) said: 

With regard to doing sa‘i after tawaaf; is it stipulated that
it must be preceded by tawaaf? 

Answer: Yes, that is stipulated. If a person starts by doing
sa‘i before tawaaf, he must repeat it after tawaaf, because he did it in the
wrong sequence. 

If someone were to ask: What do you say about the saheeh
report which says that the Messenger of Allah (blessings and peace of Allah
be upon him) was asked a question by a man who said to him: I did sa‘i
before tawaaf, and he said: “Do it and it does not matter?” The answer is
that this was in the case of Hajj, not ‘umrah. 

If it is said: What applies to Hajj also applies to ‘umrah
unless there is evidence to the contrary, because tawaaf and sa‘i are both
essential parts of both Hajj and ‘Umrah? 

Our response is to say: This analogy is not sound, because
doing things in the wrong sequence in ‘umrah is a very serious mistake. That
is because ‘umrah involves nothing but tawaaf, sa‘i, and shaving the head or
cutting the hair, but doing things in the wrong order in Hajj does not
affect it at all, because in Hajj five rituals are done in a single day. So
it is not valid to draw an analogy between ‘umrah and Hajj in this regard.
It was narrated from ‘Ata’ ibn Abi Rabaah (may Allah have mercy on him), the
scholar of Makkah, that he regarded it as permissible to do sa‘i before
tawaaf in ‘umrah, and this was also stated by some of the scholars. 

Some of the scholars were of the view that this is
permissible if one forgets or is unaware, but is not permissible if one
knows and remembers the ruling. 

End quote from ash-Sharh al-Mumti‘ (7/273) 

Shaykh Ibn Baaz (may Allah have mercy on him) was of the view
that it is valid to do sa‘i before tawaaf in the case of ‘umrah, as in the
case of Hajj. 

He (may Allah have mercy on him) said: It is proven from the
Prophet (blessings and peace of Allah be upon him) that he said during the
Farewell Pilgrimage, when he was asked about the actions of the Day of
Sacrifice – namely stoning the jamaraat, offering the sacrifice, shaving the
head or cutting the hair, tawaaf and sa‘i – and doing some of them before
others – he said: “It does not matter.” 

This general answer includes doing sa‘i before tawaaf in both
Hajj and ‘umrah. This was also the view of a number of scholars, and it is
supported by the report narrated by Abu Dawood with a saheeh isnaad from
Usaamah ibn Shurayk, according to which the Prophet (blessings and peace of
Allah be upon him) was asked about someone who did sa‘i before tawaaf, and
he said: “It does not matter.” This response includes the sa‘i of both Hajj
and ‘umrah, and there is nothing in the clear saheeh evidence to indicate
otherwise. But it is prescribed to repeat it after the tawaaf of Hajj or
‘umrah, so as to be on the safe side, and so as to avoid an area concerning
which the scholars differed, and also so as to follow what the Prophet
(blessings and peace of Allah be upon him) did during his Hajj and ‘umrah. 

It may be that what Shaykh Taqiy ad-Deen (may Allah have
mercy on him) said about sa‘i coming after tawaaf being the subject of
consensus, was that this is what is preferable. As for whether that is
permissible, there is a difference of opinion concerning that, as we have
indicated. Among those who stated that clearly was the author of al-Mughni
(3/390) when he (may Allah have mercy on him) narrated from ‘Ata’ that it is
permissible in all cases, and according to one of the two reports narrated
from Ahmad, it is permissible in the case of the one who forgot.

End quote from Fataawa ash-Shaykh Ibn Baaz (17/339) 

He (may Allah have mercy on him) was asked: Is it permissible
to do sa‘i before tawaaf, whether that is in Hajj or in ‘umrah? 

He replied:

The Sunnah is to do tawaaf first, then sa‘i after that. If a
person does sa‘i before tawaaf out of ignorance, there is no blame on him
for that. It is proven that a man asked the Prophet (blessings and peace of
Allah be upon him): I did sa‘i before I did tawaaf. He said: “It does not
matter.” This indicates that if a person does sa‘i first, that is
acceptable, but the Sunnah is to do tawaaf and then do sa‘i. This is the
Sunnah in both ‘umrah and Hajj.

End quote from Fataawa Ibn Baaz (17/337). 

Based on that, the one who does sa i before tawaaf out of
ignorance may be excused. 

With regard to your cutting your hair before doing, tawaaf
this is prohibited, as mentioned above, but you do not have to offer a
compensatory sacrifice, because you were unaware of the ruling, but you have
to cut your hair now. 

If you can go back to Makkah and do tawaaf, then do sa‘i,
then exit ihram after that and cut your hair, then this is better and is
more on the safe side, so that you can exit your ihram for certain, and do
your ‘umrah in the best manner. 

If you are not able to do that, then cut your hair now, and
your ‘umrah will be valid, in sha Allah. 

And Allah knows best.