Hi,
1) This is a known problem when you have an old bike without proper maintenance. I think the chain might be stretched, it causes the chain to jump over the rear sprockets under hard load. Measure your chain extension - every link must be exactly 1/2 inch. If the chain is more than 1% extended (measure something about 15-30 chainlinks), it needs replacement. Most probably your rear sprocket cassette will need replacement too, because a stretched chain will deteriorate the cassette quickly. In the future you will need to do chain maintenance (cleaning and lubrication) every 100 to 200 miles to make it last. There is a lot of information about chain maintenance from Sheldon Brown.
2) If you have doubts about the correct chain length, put the chain on both largest rear sprocket and largest front chainring, you shoud have about 2 free links.