Was it controlled by the Republican government? Or was it basically like an independent country (or multiple independent ones)?

The Basque territory had a fair bit of autonomy under the leadership of President Jose Antonio Aguirre. It was short-lived (under two years) so there wasn't a lot of time for the regionalism/nationalism tension to find an equilibrium--especially given that there was a war underway. I know less about Asturias and the rest of northern Spain that began the war under mostly Republican control, however.

How well were the armies in the North organized and equipped?

Few if any units in Republican territory could be described as lavishly equipped or even well equipped. So the reality of the armies in the northern Republican zones being under equipped is hardly an aberration. They did have a couple arms and ammunition manufacturers in the Basque territories and some explosives in Asturias that were put to good use, but they were woefully short on automatic weapons and insufficiently supplied with artillery. It was also difficult to have anything approaching a standard rifle or ammunition, and complexity is the bane of logistics.

Also, were any major efforts made to connect the North to the rest of the Republic? Or if not, why not? Too strategically impractical or were there also political reasons?

There were no successful attempts to rejoin the Basque territories and the rest of northern Spain to Republican control after it was severed by Mola's advance in the north. The Republicans had a difficult time even mounting offensives for a number of reasons. They were starved for supplies, outnumbered, and largely outgunned. There was certainly a desire to reconnect with the Basques after contact was severed, but the Republicans only took the strategic initiative a few times in the war and each was quite brief and quite costly. The lack of sufficient automatic weapons and artillery support (to say nothing of deficits in fighters, bombers, logistics, armored vehicles, and more) made Republican offensives have a difficult time after their initial successes. Once the Nationalists were able to funnel troops into the area in question (which was a choice by Franco on a number of occasions, he would often ignore Republican weaknesses in other areas with the explicit purpose of grinding out a slow victory that would crush the left rather than gain a victory that he thought would be too fast to be lasting) the Nationalists were able to exploit their material advantages each time. So of your offered options I would say that "strategical impractical" is the best, especially given the high cost of each Republican offensive.