The One with the Thumb - S1-E3
Visible crew/equipment: At the end, just as Monica walks off a tape marker can be seen on the floor where she was stood. This is as the credits are on screen. (00:22:30)
The One With Joey's Porsche - S6-E5
Visible crew/equipment: When the camera follows Phoebe as she is going to clean up the vase she knocked over, the foot of the baby is still visible in the shot and you can see how someone picked it up as the foot goes straight up. Then as the camera moves back revealing there are only 2 babies, you can see a shadow of a crew member moving on the ground. (00:16:00)
The One With The Apothecary Table - S6-E11
Visible crew/equipment: Right after the "I'll Be There For You" intro, the camera shows the view from outside the "Central Perk" café. As it does so, a couple of stage lights are visible through the upper part of the window, obviously part of the set on the inside of the building.
Visible crew/equipment: Most of the time when they show the Central Perk logo on the window. If you look inside you see the set because the wall doesn't go far enough.
The One Where Paul's The Man - S6-E22
Visible crew/equipment: In the scene where Rachel is trying to hide Ross from Paul, when he's about to go into Paul's bedroom after he says "Did you really hear a bear?", look at the window behind Rachel when the shot is on her. You can see a large light reflected in the glass and it moves slightly (not just in conjunction with the camera, it moves before the camera shifts).
Visible crew/equipment: When Ross and Rachel realise they are in an empty apartment, studio lighting can be seen over their heads in the ceiling as Ross leads Rachel into the centre of the room. (00:04:35)
The One Where Chandler Doesn't Like Dogs - S7-E8
Visible crew/equipment: When Chandler sees the empty dog basket on the floor at Ross' there is a panning shot to the door. If you look carefully, you can see a crewmember wearing blue jeans and white shoes. (00:17:25)
The One with the Monkey - S1-E10
Visible crew/equipment: After Max tells Phoebe that he and David got the grant and are leaving for Minsk on January 1st, when Phoebe goes to see David, just as she knocks on the door a boom mic can be seen moving at the top of the screen, before it cuts to David's closeup. (00:12:15)
The One With The Football - S3-E9
Visible crew/equipment: Just before Monica tackles Chandler to the floor, there is a shot with a very large boom mic visible at the top of the screen (only visible in fullscreen version). (00:13:10)
The One With The Football - S3-E9
Visible crew/equipment: When Ross is explaining the playing area near the start, the boom mic once again dips into view and follows him (only visible in fullscreen version). (00:05:40)
The One With The Football - S3-E9
Visible crew/equipment: Pay close attention when we see the gang outside for the first time. In the upper left-hand corner, you can see a boom mic moving about in the shot, clear as day. (00:04:30)
The One with the Boobies - S1-E13
Visible crew/equipment: When Joey's mum pushes the sofa bed away, a crew member can be seen hiding behind the sofa. (00:17:00)
The One Where They're Up All Night - S7-E12
Visible crew/equipment: Right after Tag wonders if he can see his apartment from there, when Ross says "wow, look at all the stars" a boom-mic is visible for a short amount of time.
The One With Mac & C.H.E.E.S.E. - S6-E20
Visible crew/equipment: A stage light can be seen at the top left hand side of the screen just after Joey slams the door shut after practising a scene with Phoebe. It is visible in front of the blue picture on the wall.
The One with the Dozen Lasagnas - S1-E12
Visible crew/equipment: When Joey and Chandler walk into their apartment and the table falls over when Joey puts the key on it, the edge of the set and studio lighting are visible at the top of shot. The end of the set wall is also visible at the right side of shot. (00:04:50)
The One With Ross's Library Book - S7-E7
Visible crew/equipment: When Chandler puts a plate down next to the Chinese food, and Monica walks in a boom mic can be seen over her head. (best on DVD). (00:06:20)
The One With The Soap Opera Party - S9-E20
Visible crew/equipment: When the roof party begins, above the door a boom mic is visible.
The One With Rachel's Crush - S4-E13
Visible crew/equipment: When Monica is wafting the smell of cookies to the other in the big apartment, there is a shot of Monica, and all the stage lights are reflected in the TV screen. (00:09:55)
The One with the Monkey - S1-E10
Visible crew/equipment: When Chandler is demanding someone to kiss him, a boom mic can be seen slightly dropping into the shot. (00:24:20)
The One With The Ballroom Dancing - S4-E4
Visible crew/equipment: When Phoebe and Rick are sat on the massage table, keep an eye behind Phoebe's head when the camera is on her, particularly after Rick says, "I can always find another masseuse." You can see the shadow of the boom mic moving.
Answer: The painting doesn't have a name, and if it does it is never mentioned in this episode.