Yes. I'll give you a quick rundown of the history of the series. 

2001 - Gyakuten Saiban(Ace Attorney in the west) is released in Japan for the Gameboy Advanced. 

2002 - Gyakuten Saiban 2 is released in Japan for the Gameboy Advanced. 

2004 - Gyakuten Saiban 3 is released in Japan for the Gameboy Advanced.

2005 - Yomigaeru Gyakuten, a remake of the first game including a new case, is released in Japan for the DS. One month later it hits western shores. 

2006 - Gyakuten Saiban 2 is released worldwide for the DS. 

2007 - Gyakuten Saiban 3 is released worldwide for the DS. Gyakuten Saiban 4(Apollo Justice Ace Attorney) is released in Japan for the DS.

2008 - Apollo Justice Ace Attorney hits western shores. 

2009 - Gyakuten Kenji(Ace Attorney Investigations) is released for the DS in Japan.

2010 - Ace Attorney Investigations hits western shores.

2011 - Gyakuten Kenji 2 is released for the DS in Japan

2013 - Gyakuten Saiban 5(Dual Destinies) is released worldwide for the 3DS. 

So to answer your initial question, yes, they are all connected to each other. They are consecutive. 

Well damn. That means I have to start from the beginning when I get around to it. Where does Professor Layton vs Phoenix Wright fall on this timeline (if anywhere)?

It's non-canon.

You can start from anywhere really. The one I think is the worst place to start(Because you won't get any of the references, and it spoils the previous game) is Dual Destinies. The Apollo Justice.

I'd recommend starting from the beginning though.

Good to know. Thanks for the advice! I've always wanted to get into visual novel-type games, and this seems like a neat series to start off with.

Well it is the most popular VN.

Looking at sales, that much is obvious, lol. Not many games around to compete with that title anymore. Only Danganronpa comes to mind.

Another one I need to get around to playing. 

I'm fully expecting a trailer for SMT x FE for E3 this year.

They need to do more than just release Fire Emblem 14 for the 25th anniversary.

Same. Don't know how I would manage to play it though, seeing as how I don't have a PSP anymore and I'm not getting a Vita anytime soon

Be prepared to be disappointed :P

I, too, am hoping they release some footage though..