I'm not sure I understand your question. But I think this might help?
Zillow tracks all listings for all periods. When you re-list the property, it will restart the counter for "days listed" (assuming it is down for a reasonable period of time, I'm not sure their exact specification, I know we've been down for 24-48 hours and it did not restart). But, people will always be able to see the prior listing history under the Price History section of the listing. It will show when it was listed, at what price, and when it was delisted. I actually look at this section when I'm doing a market analysis. A property that has been on the market for months, with no price reduction, isn't a property I want to compare to. Clearly their pricing strategy isn't working for some reason.
If you are asking about taking it down because winter is slow... That depends on a lot of personal factors such as your finances. Can you afford to have zero possibility of a tenant for those months that you don't have it listed? Are you available to monitor the property during vacancy? (Ensure no trespassing/squatters/maintenance issues. I go to vacant properties at least weekly unless they are multi-family with active neighbors.)
Generally speaking, if a property isn't moving (and others in the area are), either you are priced too high or there is something lacking with your property. For example, I've had a couple of 3+ bedrooms sit for a longer period of time because they only had one parking space available (and no on-street options nearby). Almost nobody that wants a 3-bedroom has only one car, except the occasional single parent with small children, or solo professional that wants a home office and guest bedroom. That's a small pool to pull candidates from. We got a ton of leads, but the parking was the dealbreaker for almost everyone. The owners refused my advice to reduce price and they both sat vacant for months. (One went with another manager eventually, and the property is listed for $400/month less than we listed it, and the other finally got rented to a solo professional.) Are you getting leads? If not, price might be the first problem. If so, what is their feedback? Don't be afraid to ask them why they don't want to apply.
When I'm having pricing discussions with owners I work with, I always remind them: a month of vacancy costs your bottom line more than a unit filled with a rent reduction. For example: if your rent is set at $2,400 per month. One month of vacancy has cost you $2,400. If you reduce to $2,300 and are able to fill the property faster, yes, you are "losing" $100 per month, but even at that rate, it would take two whole years to equal the loss of the $2,400 vacancy month. People get stuck on "this is my bottom dollar rent" and shoot themselves in the foot refusing to make adjustments.
I hope that covers your question(s)
