IMO Kibud Av DOES apply to in-laws, especially because your spouse has that obligation,and you and he are as one unit.
No one says you have to like them, but please show them respect.
Aries is right about so much in her post. Re-read it. Someday,
G-d willng, you will be a MIL yourself and I GUARANTEE your perspective will change greatly. Nothing is more important to a grandmother than her grandchildren, and perhaps your MIL is a little hyper about some things,
BUT –
when you go out, you SHOULD be easily reachable by phone for ANY babysitter. She was not wrong. Unless she was calling every five minutes, she was reasonable in being concerned that she tried for a long time with no success, when your child was coughing badly.
If she was angry about you not giving medicine, first actually consider the notion that maybe she has a point, and if not, just say that your doctor did not want you to give him medicine unless absolutely necessary, and when you left he was not coughing, nor had he been before that. HOWEVER, you should have left medicine for her on the chance he would have a coughing attack, with instructions on dosage and a medicine spoon. That is being careful and conscientious with ANY babysitter.
If she is regularly babysitting for you and saving you a fortune in sitter costs, BE GRATEFUL or stop asking her to babysit. Simple as that. I can understand and even sympathize with your negative feelings about this situation, but in the end, you are benefiting from her chessed, and that is what I believe you should focus on, more than the negatives.
Other than that, let your husband handle his mother, and don’t let this become an issue of Shalom Bayis for you. If she was coming over to make breakfast for him every morning during Sheva Brachos week, that should have tipped you off to a future problem, and you should have made your stand then. My father-in-law O”H, whom I loved with all my heart, once commented in a little bit of a nosey way about something, and I told him (respectfully) that that was between my husband and me to decide, though I appreciated his concern. He understood, and refrained from further comment on the subject. We had a loving and wonderful relationship until his death at age 94.
Your MIL is not the enemy. She may be too outspoken for your taste (and for mine, too, btw), but she raised your husband, the man you chose to marry, so presumably she did SOMETHING right. Try to accept her with all her perceived flaws (and we ALL have those), and try not to let inconsequential issues, and they are inconsequential in the great scheme of things, prevent you from having a pleasant relationship, especially when she is doing you a favor.