As you mention, cooking the steak to medium rare does indeed kill the bacteria on the surface of the steak which is where most of the risk of contamination exists, so immediately after you've cooked the steak, if you eat it, you're probably pretty safe unless there happened to be bad organisms living inside the meat.
Assuming that you've got a good piece of meat, with nothing terrible inside it, after you cook the meat, what you need to worry about most is bacteria that you introduce to the meat, and so to answer your question, cutting meat shouldn't affect the safety of storing meat after cooking at all (as long as you use a clean knife/cutting board).
While cooking the meat does initially kill all of the surface bacteria, the fact that the surface was once cooked doesn't really do anything to deter new bacteria from moving in and going to town on the meat. What this means is that you need to be very careful to keep meat in sanitary conditions regardless of whether the exposed surface has been previously cooked or not if you're planning to eat it without cooking it again.
So, to recap, if you take a piece of meat where all bacteria has been killed via cooking, and seal it without introducing new bacteria, you should be safe to eat it.
The one remaining issue is that we're not talking about a steak that's necessarily had all bacteria killed. There's a possibility that some bacteria survived the cooking inside the meat. If this is the case, you may be safe to eat the meat right away if there are very few of these bacteria present (depending on what they are), but the longer you wait - even if you seal the whole thing up - the longer you're giving those bacteria to multiply into large enough quantities that they can destroy the meat and/or sicken you. To avoid this, you want to put the meat in the refrigerator, as soon as possible to slow bacterial action, and eat it as soon as possible. Like John, I've had success with around 3-5 days, but it really depends on what you're starting with, so I'd highly recommend that you have a good look/sniff before you eat to see if there are any signs of spoilage and discard if so. Again, this is a risk whether you cut the meat or not, so that isn't really a factor here.
Regarding partially eaten steaks, this is just introducing one more place where bacteria can get on the steak before it's sealed up and refrigerated. I could see this being no issue (if you're cutting off a piece of the steak, and sealing/refrigerating the rest while eating), or a significant issue if you're cutting the steak with utensils that have been in your mouth, or worse, trying to store a piece of steak that's been in your mouth. In those cases you're almost certainly introducing more bacteria and decreasing the amount of time you'll have before spoilage occurs.
So, in summary, cut the meat if you like - that should make no difference. Get it sealed up as soon as possible without exposing it to anything non-sterile. Put it in the refrigerator as soon as possible. Throw it out if at any point you detect spoilage. Following these steps you should be safe to keep your leftovers for 3-5 days and only rarely need to discard spoiled food.
This is a very broad question. My recommendation is to look up your specific product at
the US FDA Food Safety page.
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Best Answer
The comment of local food safety is spot on, they are the authority on legality for you and that is one of the criteria you must meet for your own protection and that of your clients/guests. As you ask specifically about health department requirements, they are the authority, so go straight to the source to satisfy them and that is more of a legal question, so outside of what we can really answer.
Bacterial growth danger zone is 40-140F, so in general the advice is usually to stay out of that zone. Now, on you specific item, raw carrots are not going to be a high illness risk I would think. However, going much into that range is also going to put you into a zone that the quality, not just the safety of the product will deteriorate more quickly if you are in that range. Personal observation is that carrot will hold up fairly well up into the 50's, but why? If you are in a salad bar setup, you likely have other items that may deteriorate more quickly so to handle them you want temps at least close to that 40F number or lower such as using an ice bath if the items will remain for an extended time. And if you have items such as those made with mayonnaise, boiled eggs or other cooked items, any meats, those must be outside that high growth zone so above 140F for hot items and below 40F or so for cold. If you need to chill or keep warm some items, you might as well do all.