Some answers, in no particular order:
We keep the laws because that's what Hashem commanded. Whether we understand or not. The overarching theme in Judaism is subservience. We serve Hashem as a slave serves their master, whether we 'feel it' or not, and whether it makes sense to us or not.
That said, our sages did give reasons for many laws. These reasons serve mainly to help us connect to the laws/commandments and keep them, rather than knowing the actual reason Hashem so commanded. It's beyond the capability of a mortal mind to know the 'thought process' if you will of G-d.
A reason given for niddah is to constantly refresh the marriage with the same love as a newlywed. After not being able to touch at all for almost 2 weeks, the sexual tension is extremely high, and each spouse longs for the other.
The very strict niddah separation laws are only for spouses. So a husband and wife won't sit next to each other on a bus while niddah, but he would sit next to his sister say, even if she's a niddah (not that he'd know).
Covering hair is part of the overall commandment of modesty for a woman to cover her parts that should be clothed, and that show off her beauty. This is anything above the elbow, below the collarbone, and above the ankle. The thigh (above the knee) must not only be covered, but also not defined, hence why trousers are prohibited.
Married or once married women must also cover their hair. Using someone else's hair as a cover (ie wig) is something of a workaround, and is indeed frowned upon by some of the more religious communities. Especially if it's indistinguishable from the real thing.
Many women only wear shorter wigs, and with hats or bands.
In Judaism we believe Hashem put us here for a reason. There are clearly defined gender roles, some codified in the law while others are more cultural. But the basic idea is that we're here for a reason, so our lives should be set out to achieve our purpose, rather than pursue individual wants and whims.
That isn't to say that people can't have their own pursuits, but overall there's a sense of duty.
Men are commanded to study Torah. Some do it as a full time thing, and in those families it'll often be the woman only who works (in addition to bursaries given by the study institution, which they raise from charity).
Other men just set aside a fixed portion of the day to study. Anything from minutes to hours. And the rest of the time they work at jobs. This is actually the majority of men (outside of Israel).
Shabbos is complex with many laws and restrictions. However, by and large, it's far more pleasurable than restrictive.
We have long family meals Friday night and Shabbos lunch, where we interact with one another without distractions. Imagine two dinner parties a week but with your own family, no phones, no TV, no running errands or work. Just chilling together, talking about your week, sharing Torah thoughts, laughing and singing together.
Men have to pray 3 times a day in the synagogue. Or rather in a quorum (minyan) of 10 adults. It's morning, afternoon and night, but the exact timings can be flexible.
Mothers will usually do at least the morning prayers, depending on how busy they are. Unmarried girls/women would often do the afternoon prayers too. These don't have to be done in synagogue.
Before and after eating or drinking we say a blessing. Also after using the toilet. We can also pray at anytime in our own words, and this is encouraged. Hashem is with us at all times.