It's been a while since I worked on climates (although I'll be starting again as I re-work my world geography) so I'm quite rusty, but I'll do my best to answer some of your questions. Overall I think it would be helpful if you could post your input maps from topography onwards for us to review; otherwise it's hard to answer some of these questions.
I'd say that the July ICTZ should move further north there; note that on Earth the ICTZ is pulled northwards by the Mesoamerican peninsula, which is even narrower. Your instinct that the continent should be more monsoon-influenced seems correct to me.
I don't have a great answer for this; certainly there should be some desert there in the subtropical latitudes, but my instinct is that there's probably a little too much at the moment. I think the continent is large enough for there to be at least some small high and low pressure zones, and thus some monsoon rains to moisten up the coast at least. My sense is that the hot desert reaches a little too close to the equator in the south; by 15º or so N or S the equatorial rains usually create at least a steppe or savanna climate. It's also hard to tell if there's any topography affecting precipitation, blocking monsoon rains and/or creating orographic lift.2) I am quite unsure about the general climate of the "long narrow" landmass of the central continent. It is largely within the subtropical ridge (20-40N). It receives a warm current on its southern side, and a cold (polar) current on its northern side. During winter, the Westerlies' influence is quite strong reaching about 30N, giving a band of Mediterranean-like climate as the Westerlies retreat northwards in summer. However: I am unsure whether it should be this dry overall: I have no idea where the winds would be blowing on the landmass in most of this region. Below 20N to the equator, the trade winds would be quite stable as the ITCZ remains around the equator here year-round. I think the land wouldn't develop any strong high or low pressure zones either. Perhaps the southern warm-current and northern cold-current would have a strong effect I haven't accounted for?
That seems more or less correct to me.3) I have given the eastern half of "long-narrow" a fully-humid subtropical climate, as it receives poleward winds from the high-pressure zone to the east, similar to North America's east coast or China. It borders a patch of Mediterranean climate, as I think there would be a small high-pressure centre to the north in summer (due to the cold current meeting the hot land), but this might be unrealistic?
This is why it would be useful to see your input maps; I can't tell which way the winds are blowing here so it's hard to answer this question. If they're coming from the west, then the area should be wet. If they're coming from the inland to the east then maybe it would be a bit dryer, but I think it should be predominately west winds at this latitude, which would give the coast a monsoon climate even if there's a desert to the northwest. Note that the reason Somalia is so dry is that the Sahel and the Ethiopian Highlands block all the monsoon winds coming from the west; if anything it might be dryer in the eastern part rather than the west.4) In the tropical centre of the continent, I expect that the northern half will be overall drier, as the winds mostly blow overland from the north to the ITCZ. The winds coming from the south might be wetter, as they cross more ocean. Around 10N to the west, there is a kind of "reverse-Somalia" – I thought the winds here blow parallel to the coast, coming overland through a desert, so there is little precipitation. Perhaps the warm winds would actually blow more onshore (eastwards) here, and warm current could make this area (10-20N) wetter? (I find that these little details are what make climate so interesting.)
My instinct is that this should be more humid, or at least some of that area, but again it's a little hard to see without seeing your maps of the winds and pressure zones. Also, if there are any mountains here topography would make somewhat of a difference.5) In the South-Eastern part of the main continent, I have given it a large "winter dry" subtropical climate. In winter, I imagine most of the winds will be blowing offshore towards the ITCZ in the north. Perhaps the far-eastern coast should be "fully-humid" (like China), where the high-pressure meets wet air from the ocean even in winter?
That seems correct to me, though I'm not 100% sure; the current would be the main part of this that would have any influence.6) I'm unsure if I've properly factored in how the "triangle-shaped" body of water would affect climate here? I imagine there'd more or less be a semi-permanent high due to the cold current, so I've made both sides quite arid.
It looks great overall by the way! Love the land shapes. My only other minor correction is that the islands in the far east of the eastern continent should be tropical rainforest at that latitude.Any advice much appreciated. Thank you!