Matt LeBlanc had a fever and flu the first day. He was at 102 degrees during the pilot and had to repeatedly use a variety of remedies to attempt to cure it.
When Monica goes to get Ross his coffee, a very brief shot of Rachel can be seen passing the doors to the coffeehouse, presumably on the way to Monica's apartment. However, one must look very carefully to see this.
NBC executives were worried that the content of the show was too racy and that the audience would dislike Monica for sleeping with Paul the Wine Guy on their first date. After the taping of the pilot, the studio audience was given a questionnaire regarding Monica's promiscuity by NBC, and the survey results indicated NBC was worried for nothing, as the audience didn't care.
Joey and Chandler pour their Rachel-made coffee into the plant on the table. At the end of the episode, the plant on the table is dead, likely as a result of Joey and Chandler pouring coffee in it.
Right here in the first episode, Phoebe, played by Lisa Kudrow, mentions having a twin sister. So, Ursula doesn't just later on appear out of the blue. The full story is, Lisa Kudrow previously appeared as waitress Ursula in Mad About You (1992) and this was all just a funny twist thought up by the producers as Lisa continued to appear in that show as well until 1999.