Can I sell the stocks acquired through call to open option in any quantity and at any time or do I have to sell through (put) options only? Also, are stocks purchased through option eligible for dividends and splits?
Once the call has been exercised and you have bought the shares at the strike price, it's no different than if you had purchased them on the open market (just at a different price). You can sell them, are eligible for dividends, splits, etc., and can buy protective puts or sell covered calls on them.
While the call is still open (meaning you have bought a call but have not exercised it yet) you do not actually own the stock, so you cannot sell them or get dividends.
In the case of splits or special dividends, though, the options contracts are adjusted to reflect the new equivalent value. So if you own call options on a stock that splits 2:1, your option contract will be adjusted to reflect an option for twice as many shares at half the strike price (adding a cash component to account for any rounding difference). That way, call option holders aren't screwed over if a company splits and the stock price drops in half.
Also note that buying a put in not the opposite of buying a call. Buying a call and buying a put creates a "straddle" (or "strangle" if they are at different strikes), where you profit if the stock moves significantly up or down away from the strike price(s).
To cancel out a bought call option you would instead sell an equivalent call option (same strike and maturity).
You can do anything you want with the stock you receive from from exercising an option - including selling it or collecting dividends. However, there are several reasons why you probably don't want to exercise before expiration. It's usually best to execute an offsetting transaction (as D Stanley pointed out, selling a equivalent call) to close your position. This avoids throwing away the time value remaining on your option, incurring additional risk, and tying up cash.