How to estimate density of Polygon intersections in R? - r

I have many polygons that represent search areas of different people. By intersecting all that areas I want to get density map - 1 person searched in this area, two in that area and so on.
My trouble is that I have >5k geoJSON polygons and I need to intersect all of them. Is there a way to do it R's sf package or (less preferably) in ArcGIS?

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Determining landcover %'s (polygon layer) that is not within burned areas (raster layer)

I am trying to extract information on the % of different vegetation cover types within areas that have not been burned. However, the data I was given for the park includes areas that have been burned as a raster file in NAD_83 projection, and the vegetation cover as a polygon layer in WGS_84 projection. Essentially, I'm trying to erase the overlap between areas that have been burned and the vegetation layer to only look at vegetation cover types in areas that have not been burned. The vegetation is broken into 12 class labels based on understory/shrub cover type, and I want a percentage of how much each cover type occurs in unburned area. Can anyone help with this transformation?
I have access to ArcGIS Desktop v 10.8.2 with an Advanced licence.
I have tried converting the raster to points which worked, but I don't know how to make the points into polygons that perfectly match the size/shape of the original raster. I don't know another way to "erase" the two layers other than by trying to convert the raster to a polygon with the same coordinate system.
ArcMap is probably reprojecting the data on the fly, but you could start by transforming the raster and vector layers to a common datum.
If I understand correct, you want to know the area of veg types represented as polygons that are within unburned areas which are represented as pixels in a raster.
If that is correct, I would convert the unburned areas to a polygon layer. It might help to convert the burn raster such that unburned = 1 all else = 0 or NA, then convert unburned to a polygon.
With the unburned areas represented as polygons, intersect this layer with the veg layer. Doing so will cut the veg layer by the unburn polygon layer. The result should allow you to only select polygons that intersected with the unburned layer once it is vectorized. Get the area of these polygons grouped by veg type.

R: Species distribution modeling for a riverine species - all coordinates not falling in raster cells

I am running Species Distribution Modeling in R in biomod2 package for an riverine species in the Ganges River basin so I clipped the bioclim layer with the river network that I obtained from hydrosheds. Resolution of bioclim layer and river network at 1km*1km. But the problem arose as not all coordinates fell on the raster cells (probable causes - coordinates taken at banks, river networks not correctly delineated etc.)
So how do I overcome this problem? Do I pull the coordinates to the nearest raster cell (nearest in terms of vertices? - If this is the correct method than an easier way to do this too) or do I just leave the coordinates - more than half occurrence points would be deleted.
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Is there a way to combine a bunch of coordinates in R to form a polygon? Can we also find the outline of the polygon?

I have a bunch of (x,y) coordinates that I obtained by thresholding a 3D plot. When I plot the coordinates I get this surface:
These points are densely packed (the area in the middle is not colour fill, it's all points) and have an irregular shape. How do I obtain the boundary of this surface?
I tried saving the points as a SpatialPolygon object but that included the points in the middle. I want to only extract the boundary somehow and save that as a polygon.
Can I save these points as SpatialPoints and somehow dissolve the points in the middle?

How to calculate area of polygons within distance of other polygons

I have two sf polygon objects - one of land parcels and the other of parks. All parcels are within 2 miles of at least one park, but some parcels are within that distance to multiple parks. Parks and parcels are not of uniform size or shape.
For each land parcel, I want to calculate the area of park polygons within a half mile. Other posts answer how to calculate the area of each park and the distance between polygons, but not how I can calculate the area of parks within a half mile of each parcel. Any advice would be appreciated. Thank you!
Figured I should post the steps I took to solve this:
Create a buffer sf object for the parcels (dist = 1/2 mile)
Create an intersection sf object for the buffer and the parks (this will still have the parcel IDs, since it's based on each parcel's buffer)
Use st_area the calculate the area of those intersections
convert the intersection sf object to a data.table
merge that data.table with the original parcel sf object based on PID
That results in the parcel sf object having the area of the parks within a half mile of them.

What does "region" and "polygon" mean in rasterize function from raster library in R?

When I use the function rasterize from the raster library in R, this message appears in the console:
"Found 6085 region(s) and 6409 polygon(s)".
There is 6085 polygons in the shapefile, so I was wondering what does these two terms mean ?
A polygon is a set of points (vertices) connected to each other in a certain way by lines (edges). A region is one abstraction level higher in that sense that a region consists of one or more polygons.

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