Yes, Islam does promote gender
stereotypes roles. In some cases, adhering to your gender's role is enforceable under sharia law.
Women imitating men and vice versa
The most obvious example is that women imitating men and vice versa is considered a sin in Islam. It's listed as major sin 33 by Imam Shamsu ed-Deen Dhahabi in Al-Kaba'ir (translated) or Major Sins (pdf). The main evidence is the following hadith (among others):
Narrated Ibn 'Abbas: The Prophet (ﷺ) cursed effeminate men (those men who are in the similitude (assume the manners of women) and those women who assume the manners of men, and he said, "Turn them out of your houses ." The Prophet (ﷺ) turned out such-and-such man, and `Umar turned out such-and-such woman. -- Sahih al-Bukhari 5886 (sunnah.com)
Al-Dhahabi even goes so far as to say:
When woman wears a man's clothing, she will incur Allah's curse upon herself and her husband as long as he does not rebuke her or try to discipline her character.
As an example of how this is enforced, under Malaysian sharia law, 2013 (sourced from agc.gov.my; pdf):
Any male person who, in any public place, wears a woman's attire and poses as a woman for immoral purposes shall be guilty of an offence and shall on conviction be liable to a fine not exceeding one thousand ringgit or to imprisonment for a term not exceeding one year or to both.
There's many other imbalances which promote gender roles in various ways; some examples are:
Men are the protectors of women (protector gender role):
... But the men have a degree over them [in responsibility and authority]. ... -- Qur'an 2:228
Hijab is different between men and women. Unlike men, women are viewed as requiring protection from harassment in the form of covering.
O Prophet, tell your wives and your daughters and the women of the believers to bring down over themselves [part] of their outer garments. That is more suitable that they will be known and not be abused. And ever is Allah Forgiving and Merciful. -- Qur'an 33:59
(The OP's item 2.) Men are required to provide financially for their wives (breadwinner); see Islam Q&A. (Both the husband and wife have certain rights and responsibilities; this is just one example.)
Some of these are enforced under sharia; they are enforced in varying ways depending on the region it's implemented in and the Islamic school of thought.
(The OP's item 1 is not directly part of Islam, as far as I'm aware. However, it may arise as a side-effect from Muslim women being expected to obey their parents and husbands.)