I may be able to answer a few of these questions. I'd like to try, since Prince Henry is undoubtedly one of the most interesting characters of the 15th century.

Brief early life of Prince Henry

Prince Henry was born in 1394, fifth son of King John I of Portugal and Filippa of Lancaster. Two of his siblings died in childhood, that he became the third eldest. Through his mothers side he was grandson of John of Gaunt and great-grandson of King Edward III of England. At age 19, he partook in the Portuguese conquest of the Moroccan city of Ceuta ahead of a small contingent of men, along with his brothers and his father, following which he was knighted and granted by his father the title of Duke of Viseu and later lordship of Covilhã, where he held a considerable amount of land and towns.

It is interesting to note that both these places are in the moutainous countryside of central Portugal; Although prince Henry is nowadays pictured as a "science man" (which in a way he was), he was a medieval baron first and foremost. What brought him into the path that would eventually give him the opportunity to basically kickstart the Discovery Age, was the fact that in February 1416 King John nominated him "governor" of Ceuta. Unlike what the name suggests, he was basically in charge of provisioning and administration, the actual garrison of the town being under the command of the captain-major of Ceuta (Dom Pedro de Meneses, whose father had chosen to support the Castillian side at the Battle of Aljubarrota in 1385, so we suspect he volunteered for the job as a means to rehabilitate his household).

In 1421, Prince Henry was chosen by King John I, with authorization from the Pope, to succeed as the next head-master of the powerful military Order of Christ, spiritual successor of the Templars in Portugal, that had fiefs across the Portuguese Extremadura, the Tagus river valley, Beira Baixa and a port town in southern Portugal by the border with Spain, Castro Marim. Thus he had considerable means at his disposal not only to adequately supply the garrison at Ceuta, but also to finance his expeditions. Prince Henry was a very able administrator, and undertook important developments on the town and monastery of Tomar, headquarters of the Knights of Christ. This is the reason why the caravels had large crosses of the Order painted over their sails.

It is not true that immediately after the conquest of Ceuta, or the siege that succeeded it, Prince Henry decided to build his home on the Cape of Sagres right away. He actually spent most of his life roaming between his domains throughout Portugal managing what he owned like a typical land-owning nobleman, the Algarve (southernmost Portugal) being only one of his stay-places, due to it's proximity to Ceuta. It was only after the disastrous Portuguese attempt at sieging the Moroccan city of Tangiers under his own leadership in 1437, that henceforth he sough more and more refuge from courtly affairs in the isolation of the Algarve.

Not completely related to his house precisely, but I figured I might as well take the chance to offer some context to his personal life in case any reader is interested while we're at it.

Lagos

Throughout these times the town of Lagos was his place of stay, where fishermen, chart-makers and fortune-seekers sought employ in the service of the Prince. One of them was the Venetian Alvise de Cadamosto, whom Henry entrusted a caravel to sail to Guinea and later wrote a memoir. From here the first caravels launched into the unknown.

A university professor once told me that "the "school" of Prince Henry were actually the taverns of Lagos". This is completely anecdotal on my part, but it fits the discussion around the "school of Sagres" perfectly. We could say there was indeed a "practical school" - in the sense that Henry's sponsorship is responsible for gathering and forming a generation of chart-makers, navigators, practical sea-going men etc - but a "school" in our modern sense of passing down knowledge in an institutionalized way, not at all.

What does he mean by "opposite Peninsula," and is Vila do Infante a separate place from Lagos or Sagres?

Yes, the Cape of Sagres is very much detached from Lagos, about 20km from it, you can see that in Google Maps. There are also two capes, Sagres and St. Vincent. Henry's manor was built on Sagres where a fortress now exists.

The Villa of the Prince

The Portuguese term "vila" can mean both "town" as well as "manor".

It was only after the Crown donated (by Henry's request) the land around the Cape of Sagres to him in 1443 that he began building his manor perched on top of it. It started in 1446. What exactly attracted him to a completely deserted spit of land in southwestern Portugal we don't really know. The Cape had been sacred in ancient times, dedicated to a local marine cult even, but it was deserted by the 15th century because of Muslim pirate raids.

A very small village called "Terçanabal" existed nearby, but it was abandoned. By building his manor there, Henry repopulated it; it remained a very small thing even by 15th century standards, but it was just enough to service his manor and offer shelter to passing by merchants or fishermen who sought refuge from storms within the harbours formed between the cliffs of southwestern Algarve. Henry occasionally referred to his manor in personal letters as "my manor at Terçanabal", but became henceforth known as "the Princes Town".

We don't know what his house looked like. The oldest representation of Sagres and its fort was drawn by the famous English privateer Francis Drake in 1587... because he happened to sack the place on the occasion.

Here's an image I've found that focuses clearly on the building. It shows a long, single storey house serviced by many chimneys and entrances, with a much older medieval tower adjacent. The style is consistent with southern-Portuguese architecture (single-storey, white-washed etc); I've read that it served as the garrison quarters by then, but it may simply have been adapted to that function. Here's a modern fotograph of it sometime in the 90s.

I know that Gil Eanes set out to round Cape Bojador in 1433. Was the fort built yet at that point?

During Prince Henry's time only a rudimentary fort existed as defence against pirates. The current fortress dates from the Iberian Union.

I hope this all helps.

The main biography of Prince Henry in English remains P. E. Russell's 1960 Prince Henry the Navigator: A Life. I've used a 2009 Portuguese biography entitled Henrique, o Infante by the Portuguese academician João Paulo Oliveira e Costa.