Investopedia explains how a stock split impacts the stock's options:
Each option contract is typically in control of 100 shares of an underlying security at a predetermined strike price. To find the new coverage of the option, take the split ratio and multiply by the old coverage (normally 100 shares). To find the new strike price, take the old strike price and divide by the split ratio. Say, for example, you own a call for 100 shares of XYZ with a strike price of $75. Now, if XYZ had a stock split of 2 for 1, then the option would now be for 200 shares with a strike price of $37.50. If, on the other hand, the stock split was 3 for 2, then the option would be for 150 shares with a strike price of $50.
So, yes, a 2 for 1 stock split would halve the option strike prices. Also, in case the Investopedia article isn't clear, after a split the options still control 100 shares per contract.
Regarding how a dividend affects option prices, I found an article with a good explanation:
As mentioned above, dividends payment could reduce the price of a stock due to reduction of the company's assets. It becomes intuitive to know that if a stock is expected to go down, its call options will drop in extrinsic value while its put options will gain in extrinsic value before it happens. Indeed, dividends deflate the extrinsic value of call options and inflate the extrinsic value of put options weeks or even months before an expected dividend payment.
Extrinsic value of Call Options are deflated due to dividends not only because of an expected reduction in the price of the stock but also due to the fact that call options buyers do not get paid the dividends that the stock buyers do. This makes call options of dividend paying stocks less attractive to own than the stocks itself, thereby depressing its extrinsic value. How much the value of call options drop due to dividends is really a function of its moneyness. In the money call options with high delta would be expected to drop the most on ex-date while out of the money call options with lower delta would be least affected.
If a stock is expected to drop by a certain amount, that drop would already have been priced into the extrinsic value of its put options way beforehand. This is what happens to put options of dividend paying stocks. This effect is again a function of options moneyness but this time, in the money put options raise in extrinsic value more than out of the money put options. This is because in the money put options with delta of close to -1 would gain almost dollar or dollar on the drop of a stock. As such, in the money put options would rise in extrinsic value almost as much as the dividend rate itself while out of the money put options may not experience any changes since the dividend effect may not be strong enough to bring the stock down to take those out of the money put options in the money.
So, no, a dividend of $1 will not necessarily decrease an option's price by $1 on the ex-dividend date. It depends on whether it's a call or put option, and whether the option is "in the money" or "out of the money" and by how much.