See Attire: Man's Head Covering for Prayers or Torah Study.

Wearing a mesh head covering (kipa) is OK if the threads cover more area than the spaces. 

It is customary for men to wear a head covering always, even indoors.  This is a custom, not a halacha. When praying or saying blessings or studying Torah or when in a synagogue or bet midrash, men must wear a head covering.

Wearing a hat for prayer (for men) is formal wear that shows honor to God. Men do not need to wear a hat but must have some type of head covering when saying blessings, when praying, or when studying holy texts (this is halacha). If a man said a blessing or prayer without a head covering, b'di'avad, it is OK and he does not need to repeat the blessing or prayer.     
 

As on Jewish festivals, any adult Jew, male or female, may say kiddush for him/herself and also include any other Jews of any age or gender.

Any adult Jew, male or female, may say kiddush for him/herself and, as long as he/she still needs to say kiddush for him/herself, may include any other Jews of any age or gender.
Any Jewish male, 13 years old or older, may say kiddush for anyone else, either gender and any age, even if he has already fulfilled his personal requirement of saying kiddush.
Any Jewish female, 12 years old or older, may say kiddush for any other females but not for men, except that on the 2 Passover seder nights, a Jewish female who is at least 12 years old may even say kiddush for men, if the men are not able to say it for themselves. (Women may also say kiddush for men on Shabbat evening).
Reason Any person who is obligated to fulfill the mitzva of kiddush may say it for another person.  It is questionable whether women are obligated to say (or have said for them) Jewish festival morning kiddush.
 

Minimum head covering (kipa) size should be large enough to be considered a head covering:  a 3-inch diameter would be reasonable.

You may use your sleeve or someone else's hand (but not your own hand) to cover your head to say a blessing if you are not wearing a head covering.

A man's wife does not own the lulav with him (and a wife's husband does not own her lulav), but it is assumed that each gives their lulav to the other (on the first and second day of Sukkot) as a gift with a condition that the recipient will give the lulav back to the other spouse as a gift once finished.
 

Once a man has been married, he must wear a talit when saying shacharit and musaf, even if he becomes widowed or divorced.

Once a man is married and wears a talit gadol, he stops saying the blessing on tzitzit on his talit katan; it is covered by the blessing on his talit gadol.
 

A man may not pray within visible distance of an immodestly dressed woman.  She must be far enough away that he cannot tell what she is wearing.  Alternatively, he can turn away from her.
 

It is not recommended for a man to walk between two standing women or for a woman to walk between two standing men, but either gender may walk along with two or more people of the opposite gender if all are walking.

The general rule for yichud is that a man and a woman who is not his wife or a woman and man who is not her husband may not be secluded together. If another adult can enter the room at any time without knocking, there is no problem with yichud, even if the adult is not present initially. Yichud does not apply whenever:

• A woman has a lockable door that only she controls, which is locked (in this case, men are permitted elsewhere in the building).
• Direct descendants or ancestors are in a room together (in this case, all other men/women combinations are permitted--even if the other people are not related).
• Three (or more) women and one man (or more) are in a room, except when they will be sleeping. At those times, four (or more) women and one man (or more) are permitted in a room or enclosed area.
• Two (or more) men and one (or more) woman/women are in a room.
• Other people have keys to the room and may enter at any time.
• One (married) woman is with one or more men and the woman's husband is in the same area (RMH looks at the local business district as the “area,” so wherever businesses would be advertising or marketing would be a local area).
• For other cases, consult a rabbi.