We know that Arthur was born on the same day his mother died, which triggered the Great Purge. 1x01 (The Dragon's Call) tells us it's been 20 years since the Great Purge, and 1x09 (Excalibur) is when Arthur comes of age to become Crown Prince, commonly assumed to be when he turned 21 (it might actually be 22 depending on how long the Purge lasted, but hey, 21 is a more common "coming of age" number than 22.)
Arthur's Age [Bradley James's Age During Filming]
Season 1: 20-21 
Season 2: 21-22 
Season 3: 23-24 
Season 4: 25-26 
Season 5: 29-30 
Now, his actual birthday date is a bit more complicated. 4x03 (The Wicked Day) shows that his father dies pretty soon after Arthur's birthday, and his coronation follows immediately after. Jump forwards to 5x03 (The Death Song of Uther Pendragon) and things start to get more ambiguous. When describing the Horn of Cathbhadh, Gaius says that the High Priestesses had a ritual at the Great Stones of Nemeton at Beltane, which is a festival that occurs on May 1. Now, I'm not sure if Gaius meant that the Horn is only usable during Beltane, or if the tradition just happened to take place at Beltane. But since Arthur traveled to the Stones of Nemeton to use the Horn, I'm assuming that it had to be Beltane as well.
A general summary of 5x03 goes like this: 1) Acquisition of the horn, 2) the coronation anniversary feast, 3) Arthur sets off to the Stones and uses the horn.
So Arthur's birthday + three days (his coronation) + travel time between Camelot and the Stones (~4 days maybe?) = Beltane.
Therefore, Arthur's birthday is around April 24.
Merlin is provably younger than Arthur, but the exact amount is pretty up in the air. What we know is: Ygraine's death lead to the Great Purge, and the Great Purge "ended" with the capture of the Great Dragon [1x01], which is about the time Balinor had to flee from Camelot. All this happened in less than a year (for Arthur to be turning 21 in 1x09.)
Then we have to allow time for Balinor to flee to Ealdor, become close with Hunith, and for the 9 months before Merlin was born. So at minimum, Merlin is about 1 year younger than Arthur.
Another hint we have at Merlin's age is that Kilgharrah's opening narration changed from "young boy" in season 3 to "young man" in season 4. If we interpret this as Merlin's coming of age (21) in the gap between seasons 3 and 4, then we have the following.
Merlin's Age [Colin Morgan's Age During Filming]
Season 1: 16-18 
Season 2: 17-19 
Season 3: 19-21 
Season 4: 21-23 
Season 5: 25-27 
(More leeway in age than Arthur, but this is the general idea anyways.)
The exact day of Merlin's birthday is impossible to pin down because it's never mentioned in any way in the show. Also, medieval peasants tended not to celebrate birthdays in the first place, so who knows when Merlin's is? If it's any interest to you, my headcanon says that Merlin's birthday is exactly 6 months after Arthur's because 1) it maintains their "two sides of the same coin"-ness and 2) it places his birthday a week before Samhain, an important holiday in paganism/"The Old Religion".
(Headcanon) Merlin's birthday is around October 24.
The only time Gaius's age is ever mentioned is in 3x09 (Love in the Time of Dragons). Alice gave Gaius a book for his 50th birthday, which was before the Purge. Season 3 takes place 23 years after the Purge, which places Gaius at at least 73 during season 3.
Gaius's Age [Richard Wilson's Age During Filming]
Season 1: 70 
Season 2: 71 
Season 3: 73 
Season 4: 75 
Season 5: 79 
As a side note, people often quote the factoid that "Life expectancy was only around 35 in the Middle Ages". This is somewhat misleading. Life expectancy was only that low because of the ridiculously high infant mortality rates. Once someone lived through childhood, their life expectancy was actually around 65. Even with that taken into account though, Gaius is relatively old for the time period.
Once again, nothing about Gaius's birthday. Make up a date.
There's not enough data from the show to explicitly pin down Morgana's age. We know that she was 10 when she started living at Camelot [1x12], but that's about it. The main question we have to consider is whether or not she's older than Arthur. Personally, I think she is. Her interactions with Arthur always seemed more older sister/younger brother than older brother/younger sister, but I'm sure different people have different interpretations. Also, if Morgana is older than Arthur, that means Uther cheated on Ygraine when she was still alive; I think that's in-character for Uther, but some people might think otherwise. Additionally, if Morgana is older than Arthur, that gives her claim to the crown just a tad bit more validity (elsewise, she'd be a second-born female illegitimate child, rather than a first-born female illegitimate child). I don't write the rules though, so YMMV.
All-in-all though, I can't imagine Morgana being too much older or too much younger than Arthur either way. A year or two at most.
Morgana's Age [Katie McGrath's Age During Filming]
Season 1: 19-22 
Season 2: 20-23 
Season 3: 22-25 
Season 4: 24-27 
Season 5: 28-31 
As for her birthday, we know that it happens during 3x03 (The Crystal Cave), so it's sometime in the spring/summer (because that's when all the filming for the show happens). Other than that, no data. Make up a date!