An Air Carrier pilot reported they rejected takeoff due to an aircraft landing on an intersecting runway. Tower instructed the crew to use a taxiway that was occupied with another aircraft. After airborne TRACON did not respond to their request for a weather deviation and they encountered severe turbulence.

Narrative

After getting into position on RWXX; radar showed heavy precipitation along ZZZZZ to ZZZZZ1. Visually the cumulonimbus clouds appeared further away; however the level of the rain on scope led me to request a hold for takeoff which at that moment is when the tower cleared us for takeoff. FO (First Officer) began to push thrust levers forward; and simultaneously tower canceled takeoff clearance. Reason became apparent when a Cherokee eventually crossed our runway while taking off on RW XY. The timing made it clear it would have been either a near miss with a high speed reject; or a worst case scenario with a collision. Tower had instructed us to exit the runway at taxiway 1 and I explained an aircraft was on taxiway 1. Then received instruction to taxi forward on runway and exit at taxiway 2. I later requested a heading for departure (preferably eastward) due to the radar. Tower said he did not have anything for precipitation on his screen which aligned more with what I thought I visually saw; and I replied I am sharing what has been provided by the onboard radar. He assigned 270 and I requested something more southwest if I am remembering my request correctly. He seemed to be coordinating with ATC; and eventually asked again if 270 was okay. I accepted the heading. When entering RW XX again; the radar did not indicate any precipitation with the same settings as the first attempt for takeoff. Five degrees tilt up. After takeoff; it was clear the original radar was erroneous information. We turned to 270 and contacted departure. While departing on a 270 heading; we requested a change of heading for a large cell along our path. ATC was not responding to our radio calls. I believe we may have received a clearance to ZZZZZ; however the FO was making a statement at the same time and first portion of ATC transmission was not heard. We immediately requested confirmation clearance was for our call sign; and ATC did not respond. Multiple attempts were then made; with the frequency not at saturation; and no response. I was about to transmit we are changing our heading to avert entering a heavy cell; and ATC then transmitted clearance to ZZZZZ; stating it with especially sharp and deliberate emphasis. We turned to ZZZZZ; and the new course took us directly into another cell. While attempting to call for a new heading to avert the second cell; ATC gave us a frequency change. New controller did not immediately respond when checking in and stating the need for a new heading; it took a few attempts. While ATC responded; we entered the cell. I responded in the midst of borderline severe turbulence. ATC then stated they were there to help and asked what we needed. We then exited the cell. I stated we were now clear of the storm.