Air taxi Captain reported confusing ATC instructions; faded taxiway marking; and confusing intersections resulted in taxiway and runway incursions at MYNN.

Narrative

From the ramp; we asked for taxi clearance for takeoff. Our instructions; as I recall them; were Runway 28; taxi via Lima Alpha; cross Runway 32". We were not issued hold short instructions of any kind. As we crossed the hold short lines from Runway 32 into Taxiway Alpha; I saw the red runway identifier signs for Runway 28 ahead and to our left and continued forward on Taxiway Alpha towards those signs; as we were told to taxi to runway 28. Also; as we cleared Runway 32 and entered Taxiway Alpha; I was looking for taxiway markings and signs that marked Taxiway Hotel. We had briefed the markings on the 10-9 plate on Taxiway Hotel that showed the Runway 28 hold short lines. In the back of my head; I was not understanding how there could not be runway hold short lines on Alpha between Runway 32 and Hotel. The intersection is frankly; very confusing; the demarcation of when you enter Runway 28 from A is not clear with actual signs and taxiway/runway markings and the chart is not well-depicted. Further the red runway signs facing Hotel seem to indicate the runway area is further up and seems to be in contradiction to the chart markings. As we approached Hotel; where I thought we were still on Alpha; tower told us to hold and told us that we were on Hotel and actually on the active runway. Seeing the red Runway 28 signs in front of us and to our left; this did not make sense to us. The nose of the aircraft came to a stop just over the divergence of the taxi line from Alpha onto Hotel. In satellite imagery; I can see now the white dashed line that separates Alpha from Hotel and we were past that line. However; I do not recall being able to see any markings when we taxied over that area today. I believe they are severely faded or even missing at this point. I think there are several items that made up this runway incursion and ways future incidents could be avoided. Ground should have told us to hold short of Hotel. We would have therefore been looking for a specific place to stop instead of just rolling towards the red runway signs.We did anticipated the taxi route; but I should have paid more attention to the nagging confusion in the back of my head about the plate symbology and asked ground for clarification on where and how to stay off Runway 28 once clearing Runway 32 on Alpha. The surface markings defining the line between Alpha and Hotel need to be repainted. The 10-9 plate needs a hot spot circle at this intersection with specific notes on where Alpha ends and Hotel begins and that this line is actually a runway entry line. Better would be another symbol on the plate of runway hold short lines on Alpha at Hotel; not just the ones on Hotel (though I suspect there is a technical reason that this isn't actually a runway hold short line; despite it being thus in practice.)The company should issue a bulletin warning pilots about possible confusion at this intersection. If crossing the line between Alpha and Hotel is actually entering a runway as tower stated; there should be runway signs and hold short lines at that line; not 50 yards ahead and around the corner."