As the cited article writes correctly, we are here talking about kufr, not about shirk (shirk is kufr but not all what is kufr is shirk). Please read the linked answer first to understand the meaning of those terms.
Starting from the definition in the reference question:
Kufr is the denial of anything which is known by to be a part of the religion of Islam, i.e. the Shariah of Muhammad ﷺ, through ijma or tawatur (transmission of the words and deeds of Muhammad).
ijma: consensus or agreement of the Islamic community on a point of Islamic law)
tawatur: transmission of the words and deeds of Muhammad)
Shariah can be understood in two ways:
- Shariah is the Law of God. Ijma and fiqh are the ways of humans to find out and elaborate this Law.
- Shariah is the result of those findings and elaborations.
In the following, I use it in the first sense. Whatever understanding of the word Sharia we prefer, we must be aware that the result of those findings and elaborations are an estimation of the Law of God made by humans.
Scholars follow different schools of fiqh. Between and within those schools, quite a number of rules is undisputed, many have a majority view and one or more minority views, and some are disputed.
The article is written from a Salafiya view. Salafiya does not define itself as a school of fiqh (most Salafiya scholars refer to Hanbali fiqh). A particularity of this view is that Sharia is complete, and no human law is permitted. The author expresses this in his first point:
(someone is a kaffir, i.e. not a Muslim,)
If he issues laws and regulations other than those revealed by Allaah, because the right to issue laws belongs to Allaah alone, Who has no partner, and whoever “competes” with Him in a matter which is His alone is a mushrik, because Allaah says: “Or have they partners with Allaah (false gods), who have instituted for them a religion which Allaah has not allowed?”
The majority view in Hanafi and Maliki fiqh on this subject sounds quite similar but is different: Sharia (in the first sense of the definition) is sealed; none can be added and none can be removed or altered. Nevertheless, human law can be decreed where an additional regulation is needed, at the condition that this law does not break or supersede a law that has been qualified a Sharia ruling by all or at least a majority of scholars. In this view, it is allowed for Muslims to decree or vote for additional regulations, and those laws shall be followed (e.g. you must not drive at 150 mph on a highway saying that speed limits are nowhere mentioned in Sharia teachings). See fatwas Hanafi Maliki
However inappropriate school uniforms and free mixing (at least from a certain age) are against the large majority view of Sharia; the music classes may be counted among disputed Sharia rulings.
The situation you describe seems to be in a non-Muslim country, perhaps the US. It is not astonishing that such laws are kufr because they have been made by non-Muslim, kaffir.
Is following their rules and wearing their uniforms kufr?
It is certainly kufr if you agree to such rules, saying they are ok.
It is a sin to follow rules against Sharia, i.e. not follow the Sharia but it is not kufr as long as you do so out of necessity, against your will.
If they allow democracy are they tughut?
Tughut is originally used in the meaning "commit shirk", but you seem to mean "is it kufr to allow democracy and take part in it?".
In the Salafiya view, yes, because democracy has not been established by the prophet ﷺ, and no legislation is permitted.
Not so in the majority view in Hanafi and Maliki fiqh.
- It is allowed to make laws in subjects that need a regulation that is not explicit in Sharia, so that legislation is permissible.
- Muhammad ﷺ had consented to Jews and Christians to rule according to their own laws. Alike, it is not compulsory that stately law (which is valid for everyone who is in the territory of the state) contains all rulings of Sharia. The concept of democracy with freedom of religion is compatible with Islam because it grants the full right to every Muslim to follow Sharia law, and there are institutions where this right can be claimed. In a country where Islam is a minority, a democracy with freedom of religion granted also for minorities is the best solution to live our faith righteously, and it is legitimate to grant such rights also to minorities in Muslim countries.
It is our task to raise our voice and claim within the democracy that the public system allows respecting Sharia like clothing rules or the possibility to chose to attend separate boys and girls schools on an individual level.
I try to give a very personal, clear answer according to my view (which is at least in line with Hanafi fiqh):
- None of the aforementioned is shirk (even according to Salafya)
- It is possible to live Muslim faith in a non-muslim country although sometimes you will have to fight for it. I have been living in Switzerland for almost all of my life, where 4% are officially Muslim and many less are really practicing.
- It is allowed to vote or to engage in politics. I do so, too (I am not elected but I would not refuse to be a deputee). And it is allowed to a Muslim to give out additional laws where there is no evidence from Sharia. It is obligatory to declare that those rulings are not Sharia, so nothing is added to the divine law.
- Just as you mentioned you are living in Egypt: Who were better, the democratically elected goverment or the dictators before and after?
- In a western country, girls schools are often only available in larger cities, and residential girl's schools are often run by Catholic Christians. I (male) personally attended a mixed school in the countryside, and neither my parents nor I were aware that this was haram; still it is. If your parents plan to go to a western country, you should discuss this problem and find a decent solution. Clothing is more difficult in states based on English culture as they wear school uniforms. You should be allowed to wear Hijab but you may need to fight for it.