The most interesting thing about Theodred is who he would have surrounded himself with. As a child in Edoras, he had no real peers of his own age. His cousin Eomer was born in T.A. 2991 - thirteen years after Theodred. In fact, during his childhood, his closest kin in age would likely be his aunt: we know that Theodwyn, sister to Theoden, was born in 2963, making her fifteen when Theodred was born.
Thengel and Morwen married in 2943; their second child, Theoden, was born in 2948, a gap of five years. The gap between Eomer and Eowyn is also 4-5 years, suggesting that 'five years between children' might be the general rule for Rohan (and Gondor - there are five years between Boromir and Faramir). Given the 15 year gap between Theoden and Theodwyn, that suggests the five children were born in '43 (nine months after her parents were married), '48 (Theoden), '53, '58, '63 (Theodwyn).
Elfhild, Theodred's mother, died giving birth to him; he would not remember her. Thengel, his grandfather, king of Rohan, died when he was two, so he might just remember a grizzled old man. Interestingly, though, Morwen, Thengel's wife, was nearly 20 years younger than him; she would have been only 58 when Thengel died. Further, she was not Rohirric - she was kin to the Prince of Dol Amroth, and it has been shown that the Dol Amroth line usually lived to around 100. We know that Morwen wasn't around for the War of the Ring - or else she would have been mentioned - but she may have died only a little before (she would have been 97 in 3019).
The only other nobleman we know of, who could have been around Theodred's age, was Erkenbrand - but he was born and raised in the Hornburg, again far away from Edoras.
So we have young Theodred growing up, raised by his father, now a king, and his grandmother. His teenaged aunt grew into a beautiful woman, and when Theodred was eleven, married Eomund, Lord of Aldburg, and moved out of town. Judging by the lack of mention of them, we can suppose that his other three aunts were either dead, or simply had no (living) children - otherwise, there would later have been other candidates for the throne than Eomer. His young cousins, once they were born, lived far away in Aldburg.
There were noble children of Theodred's age, and in a similar situation to him - in Gondor, Boromir was born in 2978, making him the same age as Theodred. Faramir was five years younger - and in 2984, Ecthelion II died, making Boromir the son of the Steward. Further, in 2988, Denethor's wife died; it is hard to imagine Theodred and Boromir meeting and not bonding over their near-identical life stories. And with a Gondorian grandmother, we can easily imagine the young Crown Prince spending time in Minas Tirith.
In T.A. 3000, when Theodred was 22, Eomer 9, and Eowyn only 5, everything changed. Sauron returned to Mordor, and through the Palantiri, twisted Saruman to his alliegance. Orc raids in Rohan increased dramatically - as demonstrated by the fact that, two years later, Eomund was killed. When Theodwyn died (essentially of grief), her children (Eowyn now being seven) were brought to Edoras.
How did Theodred, now 24, handle his very young cousins? The evidence suggests that he... didn't. King Theoden was now in his mid-fifties, and grief-stricken by the death of his sister and close friend (here, we might imagine, one Grima son of Galmud was a great comfort to him). Since Theoden had been the (effective) First Marshal of the Mark, that left no effective military leader in Rohan - until Theodred stepped up. He moved out of his father's house, transferring to the Hornburg, and became Second Marshal - leader of the forces of western Rohan. In fact, he rather overstepped his bounds - Elfhelm, who was under his command at the Fords of Isen, was from the Muster of Edoras, the traditional purview of the First Marshal.
In all this, Theodred would have had to deal with Erkenbrand - the Lord of the Westfold, a capable military leader in his own right, who was now under the command of someone who was either his own age (but new-come to the West-mark) or younger than him. How well did they work together? Were they friends, or rivals, or simply superior and subordinate? It's impossible to say.
From afar, Theodred would have watched his cousin Eomer grow into a tall, proud warrior, and eventually take up his father's role as Lord of Aldburg - and, indeed, become Third Marshal of the Mark, at just as young an age as Theodred had. Stranded out in the Westfold by the constant incursions of Saruman's (though of course, no one could prove they were Saruman's) orcs, did Theodred resent his cousin's easy job, back in the safe east? Or was he too busy worrying about his father, who seemed to be turning grey ahead of his time, and every time Theodred visited was more bent, more weary, less and less like Theoden of old.
Perhaps it was Morwen's death which finally broke Theoden for good, turning him into the barely-conscious king we first see in Edoras. Perhaps, too, it was this event which finally brought Theodred and Eomer together - through mutual grief, mutual respect for each other's military prowess, and mutual recognition of the perilous state Rohan had reached. Certainly we know they did reconcile - when Theodred was slain by Saruman's treachery, he spoke his last words to Elfhelm and Grimbold:
"Let me lie here - to keep the ford until Éomer comes."