The issue seems to be that mount or losetup sees the partition table as listing the second partition's size as exceeding the actual size of the image. The first clue about this is the error in dmesg when trying the mount:
EXT4-fs (loop0): bad geometry: block count 11736 exceeds size of device (11735 blocks)
Blocks here are 1 KiB, so this number squares with the sector count I get from fdisk -l on piCore-7.0.img:
Device Boot Start End Sectors Size Id Type
piCore-7.0.img1 8192 69631 61440 30M c W95 FAT32 (LBA)
piCore-7.0.img2 69648 93119 23472 11.5M 83 Linux
Since 23472 / 2 = 11736. The total space here should be:
(8191 + 61440 + 23472) * 512 = 47668736
I'm not sure if that includes the MBR, which is trivial (512 KiB), but the image is 47676928 bytes anyway.
I tried this though, and it worked:
dd if=/dev/zero of=zero bs=512 count=4000
Creates two MiB of blank. Use more if this still doesn't work. You can probably combine that command with the next one but for simplicity:
cat zero >> piCore-7.0.img
Note the >> to add to the end, not overwrite. Now this will succeed:
mount -rv -o offset=35659776 piCore-7.0.img whatever_directory
You don't actually need the fs type here.
However, you may (or may not) be disappointed with what's inside -- because tiny core is busybox based, I believe the runtime root fs is in a .tgz file there, but I am not a busybox user, so that's as far as I go. Likely if you start digging around you'll figure it out, and again, you are modifying the image with it mounted like this.