When you talk about Dalmatia you can either mean just the thin strip of coast including the islands, or a much more broad and official definition which includes the hinterland like shown on this map. If you take the strictly 2nd definition and compare it to the map of serbian population before the war and the map of area the serbian rebels occupied you will find a considerable portion of Dalmatian hinterland had sizeable Serbian population and was in fact the core of the Serbian rebellion. Knin, which is kinda included in Dalmatia was the origin and the capital of the Krajina (serb self-proclaimed state in Croatia). There were also areas next to it which had Croatian majority who were were occupied.

But how did the islands and the coast fare?
For starters, Dalmatian main connection to the rest of the county was by a road and train track going right trough Knin, meaning the Serb populated areas. Actually the first act of the rebellion was when in August 1990 the Serbs started blockading this road by rocks and fell trees, leading croats to start dubbing it the Log revolution.
The only other direct road to the rest of Croatia meant going all the way following the coast and even that was severed when Maslenica bridge was destroyed in 1991, and ferries had to be used. This had severe economic impact on Dalmatia, hampering tourism and industry.

In 1991 Croatia declared independence, with conflict and full on war starting. From roughly September to November 1991 Croatian forces blockaded JNA (Yugoslav Army) barracks throughout the country to seize much needed weapons and prevent those weapons going to the Serbian side. Dalmatia had several major (among many smaller ones) installations in Zadar, Šibenik, Split, Ploče, and on islands Vis and Lastovo.
After independence and blockade of the barracks, the JNA from the Hinterland in turn moved in directions of Sinj, Šibenik and later Zadar to relieve the sieges of the barracks and take control of Dalmatia. To avoid dragging into some unwarranted discussion, let's just say there are conflicting views on which was the real goal of these operations: de-blocking the barracks or acquiring most territory for the Serbs. There was fighting on all directions and the JNA captured Drniš but fell short or stopped just before the others.

At the same time the JRM (Yugoslav Navy) started blockading (and few times bombarding) the Dalmatian ports. The Croatians captured sea mines and coastal artillery on the islands and used them to successfully to drive the JRM out of the inner channels of the coast to the outer islands of Vis and Lastovo, effectively ending their involvement. By early December peace treaty between Croatian forces and JNA had been reached and JNA retreated from Croatia in Janurary, leaving only Serbian forces in the Hinterland (SAO Krajina area). The JRM on islands of Vis and Lastovo remained until summer of 1992, after which they, too, retreated.

It meant the end of major offensive operations in Dalmatia until operation Storm in 1995, bar some operations by Croatian forces in 1993 around Maslenica to secure a better traffic corridor. The operation Storm deals much more with the hinterland so I will not go into details here

Until now I avoided mentioning the special case of siege of Dubrovnik, a famous town on Croatian south, with no significant Serbian population and no military value whatsoever, which was attacked by JNA forces in 1991. It was attacked from direction of Montenegro in the south, and cut of in the north of the city by forces from East Herzegovina (Serb territory) and kept in land and naval blockade until May of 1992. The Old City of Dubrovnik itself was bombed.The attack on Dubrovnik was a completely unprovoked, unwarranted, blatant aggression, justified by a truly mind-boggling propaganda.

So after this brief overview, to answer some of your direct questions:

was there much conflict over borders?

In coastal Dalmatia most of the conflict was in autumn of 1991, during which all of the major towns in Dalmatia were threatened. When the peace treaty was signed in the end of 1991 and the JNA finally retreated, the fighting more or less stopped until 1995 and operation Storm, which in turn ended the war.

were the islands insulated from the conflict

Bar the few days worth of action of the coastal artillery against the JRM Navy, the islands didn't experience direct conflict. Indirectly the threats to the major ports were threats to the islands too, as they all gravitated and were depended on the ports for supplies. They were also affected by the naval blockade in 1991 and general bad connection of Dalmatia with the rest of Croatia throughout the war.

was there much of a population mix outside the border regions

The parts of Northern Dalmatian hinterland: Knin and areas behind Zadar. Other then that no, not really.

Hvar and Vis specifically

Vis was a JNA stronghold since WWII. During SFRJ Yugoslavia it was off limits to foreigners (meaning no tourism) and had several major military installations on it. As such, also due to its position, it remained in JNA hands until it was finally abandoned in summer 1992. Hvar was uneventful, except for the engagement of the JRM navy by the coastal artillery that happened near it.

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EDIT: I see you have a Croatian surname? If you speak and read croatian it would be much easier for me too provide sources and additional information.