As of our current understanding, all items that can be freebied will do so at a rate of 33% regardless of difficulty mode. Explanation here.
Success rate to run away from battles: 60% in Normal, 70% in Breezy
Special Enemy encounter rate: 20%. This also appears to not be impacted by difficulty mode.
The following sheet is a DB of enemy data, item data, and attack data ripped by Echocolat. You can find detailed information about all of these things in here, including obscure things such as vulnerability to sheep attack. Feel free to use this to inform your research.
Here is a JSON of the frame data for Super Jump (105), Ultra Jump (107), and Star Rain (114) that Echocolat also provided. This suggests that the ultimate timing for super jumps is still 3 frames like in SNES, and Ultra Jump also tapers off at 3 frames (instead of 2 frames on SNES).
Click to expand and see the JSON data.
[
{
"_action_id": 105,
"_max_num": 100,
"_hit_count_disp_thresholds": [
0,
10,
20,
50,
100
],
"_succeed_start_frames": [
0,
10,
19,
25,
29,
33,
37,
41,
43,
44,
45,
46,
47,
48,
49
],
"_succeed_frames": [
0,
61,
31,
27,
23,
19,
15,
11,
9,
8,
7,
6,
5,
4,
3
]
},
{
"_action_id": 107,
"_max_num": 100,
"_hit_count_disp_thresholds": [
0,
10,
20,
50,
100
],
"_succeed_start_frames": [
0,
10,
19,
25,
29,
33,
37,
41,
43,
44,
45,
46,
47,
48,
49
],
"_succeed_frames": [
0,
61,
31,
27,
23,
19,
15,
11,
9,
8,
7,
6,
5,
4,
3
]
},
{
"_action_id": 114,
"_max_num": 17,
"_hit_count_disp_thresholds": [
0,
5,
9,
13,
17
],
"_succeed_start_frames": [
0,
69,
12,
16,
20,
24,
28,
32,
34,
35,
36,
37,
38,
39,
40
],
"_succeed_frames": [
0,
62,
32,
28,
24,
20,
16,
12,
10,
9,
8,
7,
6,
5,
4
]
}
]