No, the proceeds from an open short will not count towards your account equity, so you won't be able to use the proceeds to enter new positions, etc.
For example, if shorting proceeds are say $10,000 from shorting 100 shares, this amount will be held by your broker towards buying the stock back when you close the position.
On top of this, from your account equity you will also need to post additional margin to cover you (or more accurately, your broker!) in the event that the stock rises before you can cover your short position.
For an unleveraged ETF, this margin intraday might be 30% for a short (or 60% for a 2x leveraged ETF, or 90% for a 3x leveraged ETF).