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camera Working with profiles/exposure

Hello Everyone!

I am very new to filmmaking, currently I do some online courses and I have numerous questions regarding the set-up and theory, I will appreciate if community will help me to clear all the things up, as currently I feel it is a mess in my head and I am trying to make my understanding very well minded. All question I've googled and searched for a clarifications, but still I struggle to gain full understanding of the topic(s).
Trying to take my hobby as serious as possible. Sorry in advance for a long read.

I have Sony A7M3
I was taught that it is better to use external monitor instead of camera monitor for better control of exposition/colors. I understand that I can buy external monitor & stabilizer and connect it all to the camera, however as of now it is costly for me so I am using Sony Imaging Edge Remote desktop application to have a monitor to my laptop. It is connected via cable.
Q1. It looks like it is not possible to do a wireless connection to monitor video Sony vs Imaging Edge Remote, am I right? If it is not possible, than I have to use sufficiently long cable to be able to move my camera. But maybe it is an awkward setup, your comments are welcomed.
Q2. I haven't found in Imaging Edge Remote any instrument that can show me if my video is over/underexposed. Anything like histogram or zebra. Is there anything that can show me this in Imaging Edge Remote?

Many Youtube videos I've seen advise to use picture profiles (like slog) and the grade the video.
I'd like to do some footage in slog mode, but I am afraid if I took something wrong regarding slog concept - I can spoil the entire video.
Can you correct my understating please.
Q3. Slog footage will be always gray and with low contrast, it is normal, it is actually the purpose of slog to record as much information in darks & lights. The postproduction grading will adjust colors & contrast. One of the easiest/fastest ways to grade is to use LUTs (however there can be some artefacts, but for my beginners level I think it is ok to use LUTs instead of manual grading).
Q4. It is bad idea to use slog in low light scenes - there is a risk of getting very noisy footage.
Q5. I downloaded some LUTs for testing and I see there is a "rec 709" in the name of the LUT. I searched for this and found that rec709 is a standard color space.
Am I right that if I am using picture profile PP3 or PP4 (ILCE7-3M) it is rec709?
Am I right that rec709 has minor effect (with comparison to slog) on the initial footage (may be a little bit more details in darks) and is pretty close to "no profile" footage?
Am I right that LUTs are designed for a particular profile? i.e. if LUT is designed for rec709 and I will apply it on image recorded in slog, most likely footage will be still low contrast and I have to add some contrast manually. Vice versa, if LUT is designed for slog (so most likely it will add some contrast to footage) and I will apply it to rec709 I will get video with very high contrast and most likely I will need to lower it manually?

When working with the normal light, it is more or less clear how to set up exposure. More issues with low and low/high lights.
Q6. For very sunny/bright environment it is better to use ND filters, correct?
Q7. For low light for me it is much more tricky. I can manage exposure mainly with ISO and shutter. But in low light I feel that autofocus is working worse than in normal light. But I also feel that with the closed shutter it is focusing better than with the opened, is it correct? Also I will appreciate if community will share with me tips and tricks for filming in low light environment (footage like noir films)

Thanks in advance!