Why were there no actresses in Shakespeare’s plays, and what did he do to compensate for that? How did people know a play was being performed during Shakespeare’s time?
Throughout theatrical history, there were very few time periods that allowed women to act (until the late 17th century). Women acted during the Italian Renaissance, in Commedia dell 'Arte troupes, and they acted in the Morality plays of the Medieval time period, but that's about it until the mid-late 17th Century. During Shakespeare's time, in England, women were not allowed on the stage. This was primarily due to issues of morality. The two exceptions I noted above allowed women, but they were expected to have familial connections to the actors. There was not an acceptance of men and women working closely together, especially in the unsavory world of the theatre (as it was perceived).
The lack of actresses explains why so many of Shakespeare's female characters disguise themselves as boys. Certainly, the boys playing the roles would be more believable as boys than as women! It also may be part of the reason that Shakespeare (and his contemporaries) wrote plays that had MANY more male roles than female roles.
In the sixteenth century in England,since no women were allowed in the theatre, young men played the roles of women. It seems odd to us today to imagine the seductive words of Romeo being said to a young man, but the audiences of the Elizabethan age were accustomed to this arrangement;with elaborate and expensive costumes for the actors and with the aid of the English imagination, the compensation was made. In addition, the appreciation for the beauty of the language and the intriguing plots of Shakespeare mitigated many detractions from the less sophisticated theatre.
England, in fact, was the last of the European countries to accept women on stage. In 1629 a visiting company of French players gave performances at Blackfriars employing actresses. However, the women were hissed and "pippin=pelted" from the stage. The English boy actors, often members of the choir at church, were very popular, having been well trained at their schools.
To answer the second part of your question . . .
In terms of the anouncing of the plays, announcements would have been distributed - like leaflets or posters - announcing the production of a play. They would also fly a flag from the top of the playhouse to let the neighborhood know a play was to be performed that day. The color of the flag would advertise the type of play to be performed (black for tragedy, white for comedy, red for history). They would also sound a trumpet to announce the play's beginning.
If you watch the beginning of Laurence Oliver's "Henry V" you will get a good visual of this.