Automap appears to be broken. I made sure to use a PRG0 ROM. I played on Mesen. I applied the patches I used in this order, all of which MMC1.
The end result? Automap flashes between random tiles that I had previously visited. The photo below shows an example in the bottom-right corner.
This is the very last bug I was mentioning in the post here:
https://www.romhacking.net/forum/index.php?topic=29403.msg407357#msg407357Where I say the following in regards to that bug:
...I wanted to mention that, according to everything that's been done and the features/points already covered, once the MMC1 (CHR-RAM) Automap is working with the animation made by Fiskbit, I think the project would be ready for an actual release!
I am hoping to have the hack ready for the Zelda 1 35th Anniversary, which is on February 21st of this year.
So all gears running, guys! And if someone can help with the last thing to meet the desired date, it'll be more than welcomed and very thankful! 
Here's the method Bogaa did to make the Visible Secrets work with the MMC1 Animation, which is what I'm using as the base to implement the fix for Automap:
https://github.com/ShadowOne333/The-Legend-of-Zelda-Redux/blob/master/MMC1/code/gameplay/visible_secrets.asm#L164-L251
As for the Automap code itself, I believe these are the routines that need to be modified to make it work with the MMC1 Animation, taking Bogaa's implementation from the Visible Secrets as the base:
https://github.com/ShadowOne333/The-Legend-of-Zelda-Redux/blob/master/MMC1/code/gameplay/automap.asm#L560-L627
Final stretch! Just Automap CHR-RAM with animation and we should be ready for release! \o/
Automap is not fully compatible with the MMC1 animation, and that's the last remaining thing to do so I can effectively make a proper release of the hack, hopefully by February 21st for the 35th Anniversary.
As for your second point:
EDIT 2: Just tried using FCEUX 2.3.0. Same issue with Automap tiles erratically flashing, but now, the item counters flash between their respective values and ellipses, albeit at a slower rate than the Automap tiles. Also, when walking sideways, Link's walking animation doesn't trigger. Instead, he blips in and out of existence.
EDIT 3: I was up all night testing different patch combos. All I got was increasingly worse results, to the point where the entire screen was starting to glitch out.
https://imgur.com/a/gUNH6Bh
In the end, I finally got it to work........by giving up and using the MMC5 version. I don't know if this is old news, and I'm too sleep-deprived to bother checking. I just wanna go to bed.....
I got this report as well from another user, IcePenguin, when he was doing some tests on his own.
I always test on FCEUX 2.3, and the only bug I had encountered so far was the Automap flickering/blinking, but this is because Automap isn't compatible 100% with the animation for MMC1.
However, it seems IcePenguin found another bug, which is entirely and solely an issue with the Automap code, and it gives the results you describe.
The problem here is that the issues both IcePenguin and you describe can ONLY be replicated if you have no previous SRAM of the game created, so if you go load up the hacked ROM blank, with no previous save done or never opened up in the emulator before, you replicate the issue, but as soon as you reset the game, the issue seems to be gone for some reason.
Example from a fresh start, with a ROM that had no previous save:
The glitchy results:

I did find the culprit of the issue though. Seems like a "solution" I made months ago to an original Automap bug simply created yet another bug:
https://www.romhacking.net/forum/index.php?topic=29403.msg403797#msg403797The solution posted by @gzip also results in a similar issue, but way more noticeable.
I'll see what can be done to fix this, but at least the issue is now identified.
EDIT:
I think I managed to fix it.
Please, to whoever was having these issues with the MMC1 version, try the newest patch in the GitHub repo within the MMC1 folders.
As for the MMC1 Automap stuff, I haven't been able to work on any hacking stuff these past days.
I am not quite sure if it will be possible to have the Automap work with MMC1 Animation before the 21st, so the decision to release the hack by that date is still up in the air. By the look of things, I doubt it, but we'll have to see.