There is a lot to unpack here, and frankly, I'm too lazy to break it all down today. SO here are the highlights I took out...

First, I always get my dander up when a Soldier starts talking about "rights." Does this person have the right to do that, or do I have the right to do this... Your question in not about RIGHTS, but about AUTHORITIES. It seems like merely semantics, but it is not.
In answer to your question, does that NCO have the authority to chew you out and smoke you... not really. From the way it reads, that NCO is not in your Chain of Command or NCO Support Channel. (S)he absolutely has the authority and RESPONSIBILITY to make on-the spot corrections. And (possibly) if that NCO was in a position of temporary authority - say an instructor or (relevant to your post) a senior mechanic seeing you do something unsafe in the motor pool, maybe an ass-chewing and/or a smoking.
Additionally, no NCO anywhere in the Army has the authority to remove your rank. They can recommend, but only Commanders can promote or demote. (That being said, let me be VERY clear that Commanders usually listen to their NCO counterparts, especially when it comes to enlisted matters - so a 1SG can't take your rank, but if he says he will recommend or "make sure" that you lose your rank, take heed!) Some NCOs DO have authority to make you lose your position, but not SGTs. 1SGs and up are usually delegated authority to transfer Soldiers within their units by their respective Commanders. PSGs sometimes have authority to shift folks around within their Platoon. In your situation, moving you would require at LEAST the 1SG, and more likely both your 1SG and the HHC 1SG. That SGT does not have the authority to move you. That being said, if that SGT can show a good cause to the HHC Medic PSG (usually the senior enlisted medic in a BN), it COULD happen... but a lot more than just that SGT is involved.

So overall, that NCO was in the wrong (at least as I read the situation). Not for reprimanding you, but for in the manner in which (s)he did so.

Regarding all of the infractions, the only one I will really comment on is the Battery terminals. I am going to assume that you were PMCSing WITHOUT your TM. Because had you been following your Tm step-by-step and actually READING the steps, you would have seen that you need to ensure that the terminals are free of corrosion and the cable are secure. I have been retired for a few years, so I can no longer quote the exact step and wording, but it is there. So in that, regardless of who trained you or how, you were dead wrong. Conducting a PMCS without the TM is dead wrong. Conducting a PMCS WITH the TM, but failing to follow the step as described is also dead wrong. If you don't know what the step means, you ask your First Line Leader or a mechanic.

I will also say that you have identified for yourself some of your personal weaknesses. Identifying a weakness is only the first step. You owe it to yourself, your unit, and your future Soldiers to work on fixing those deficiencies.

Finally, I will leave you with this.... Stuff like this happens. If you continue on for a full career with the Army, it will happen again. I wish I could say it won't, but I refuse to lie about it. You handled it correctly (from what I can see here). Take the ass chewing, comply with the NCO, and move about your day. It may be hard to do, because you feel you have been wronged, but let it go, put it behind you, and move forward. Ultimately, you belong to your line Company. If your Line Company leadership is happy with you, and with your performance, then you are good to go. If this CONTINUES to happen, however, with HHC medics getting into your business or chewing you out for prioritizing your actual unit rather than the medical PLT, then you need to escalate this to your PSG (if you have one - I know those Company HQ positions get wonky sometimes), if not your 1SG. Hopefully PSG to PSG conversation can resolve things, and then your PSG can give you clearer guidance as to priorities and which meetings/trainings/events you attend. If not, then a 1SG to 1SG conversation is almost guaranteed to solve it - those 1SGs work with each other behind the scenes a lot more than most folks realize.

Jesus, where to start? 1. Stop Whining. If you are the Senior Medic, then everything that happens good or bad is your responsibility. An FLA is medical equipment and comes with flippin manuals. I ran a IN BN Spt Plt for 2 years as Bn Ammo Chief/ PSG. ( Up till then a Grunt). Up till then never drove, TI's, PMC any military vehicle, but with the help of my 88M and the manuals- I learned. 2. If the NCO yelled, well I guess they didn't like the idea of a Vehicle burning up and or hurting someone. 3. If they aren't in your chain of command- HHC or the line Co's ISG, then they damn well don't count except for an ass-chewing/on spot correction. You know- "Yes SGT, NO excuse SGT, Will Do SGT" They you report it to your NCO chain. 4. What you think of someone's actions of a higher rank than you is immaterial, unless was sexist/racial, etc. 5. You are an NCO, if you plan on staying in or advancing, then better get use to ass-chewings, sometimes they are right, often it's just releasing steam. A long as it doesn.t affect your NCO nor progression, then don't worry about it.

The general answer in my opinion is using threats to moderate behavior should be a last resort. The chewing may have been appropriate as it provides a lesson and motivation, however, unless you had failed to do what was directly asked. I would generally recommend against any threats as an NCO. providing guidance and counseling should be the key. Threats are usually unwarranted because if something is bad enough to warrant the threat, it more often than not should come with consequences, and those consequences will come through your chain of command. Any potential consequences might get based on a recommendation from an NCO who witnessed misconduct or negligence.

At times during my career where I had issues with a soldier who was not in my Chain of Command, I would take the time to 1st make needed on the spot corrections. Secondly, if I felt the problem warranted further action, I would talk to my counterpart NCO in their Chain of Command. If the problem was with one of my soldiers I would provide counseling (not always in writing) with a plan to fix the issue, and if a problem was brought to my attention regarding one of my soldiers I would talk to the soldier in question and provide counseling and feedback to that soldier. (note that counseling does not always need to be in writing, and also does not need to be negative) Positive counseling is far too often overlooked as a tool for motivating good soldiers.

I would suggest asking your team leader how they feel about your performance and mention without assigning any fault that another team leader had expressed concern about the issues noted. I make this recommendation in order to ensure that your team leader is aware of any potential situations where you might be targeted by an outside NCO who may seem to have taken a dislike to you. It is much easier for your team leader to look out for your interests and the interests of their team if they are aware of any such situations. and it is also relevant that you can get feedback on expectations for yourself and any changes you might need to make in the way you do things.

Focus on being diplomatic when bringing this situation to the attention of your team leader.