^ Sister that is not praying it OUT of its time. The prophet prayed all prayers within its accepted timespan, which is from the adhan call till the adhan call of the prayer afterwards (except fajr it is till sunrise. Many scholars also say Isha prayer is only timed till middle of the night.). There is no punishment or loss of reward if it is prayed after the adhan by a while. This is based on authenticated Hadeeth of Arch-Angel Gibreal coming to the prophet and asking him to pray at the beginning of time and the next day he came and asked him to pray at the end of its time, then telling him prayer is required at any time between those two limits. It is also authenticated that the prophet sometimes prayed the Duhr so close to Asr prayer that companions thought he was making JamAA (grouping) for travel, because he prayed it before Asr time by 10 minutes and then sat waiting for Asr to come in and prayed it.
There is a bidah (I seem to see that it's spread in Asia today) where people call the adhan for the prayer and then almost in the same breath call for the Iqama of the prayer itself. The prophet and companions did not do that: The adhan was to alert people outside the masjid to come to prayer and therefore it is more than natural that they wait for 15 to 20 minutes till people get alert, get dressed, make ablution and come to masjid. Also time is needed especially for dhuhr to make sure people arrive and pray the 4 rakkat sunnah.
Making the prayer immediately is a bidah and makes people lose the chance of getting to prayer, and also risks making prayer before its actual time if the caller of adhan was early, especially in Fajr where today there is a bad habit of making the Adhan in Fajr Katheb (false dawn) time against the literal order of the prophet.
What muslims should do when they make the call is to wait for 15 to 20 minutes before holding the Iqama till people have gathered for the JamaAAa prayer, which is 27 times better than single prayer.
Supporting Sources: http://islamqa.com/en/ref/65941
What is the ruling on delaying Fajr prayer until the sun has almost risen and not offering it at the proper time?
We have stated in the answer to question no. 9940
the times of the five daily prayers.
The time for Fajr prayer lasts from the onset of the true dawn until the sun starts to rise.
The Prophet (peace and blessings of Allaah be upon him) said: “The time for Fajr prayer is from the breaking of dawn, so long as the sun has not started to rise.” Narrated by Muslim, 612.
If a person offers this prayer during this time, he has prayed on time. Based on this, what is mentioned in the question about praying close to sunrise not being the right time is not correct. Rather the time for Fajr prayer lasts until the sun begins to rise.
The consensus of scholars is that going to masjid IS OBLIGATORY FOR MEN, unless it's too far to hear the Adhan then it is sunnah and a legitimate excuse for not having to go. But if a man hears the adhan closely enough, he needs to go to JamaAAa prayer in masjid, otherwise he will be earning a sin. This is proven by the prophet's hadith who ordered the blind man to still go to the masjid even though he asked to stay home considering his handicap, because he said could hear the adhan to prayer. Another hadith the prophet says: "I wish I would make the call to prayer and go to the houses of those praying in their homes and burn it down upon them!" Showing what great calamity it is for men to be praying at home when they can hear the unaided Adhan call.