Leggy plants are the result of poor light (both strength and duration) the "falling over" is probably "damping off" a fungal infection. steril soil, good air circulation, and low humidity will delay or prevent fungal growth. Broccoli is a cool crop can go outside as soon as can work ground. Egg plant set out aafter last chance of frost. Artichoke is not hardy in zone 5 & goards are best form seed (planted after last danger of frost) started where they are to grow.

So I may not have a choice but to buy lamps?

IF you want to grow from seed indoors, then yes, some sort of lights, are required. 16 hours on and 8 off or 12/12 are the standard recommendations. Those who claim they can do it just using a window are misleading themselves or are content with far less than ideal transplants.

Most of us use inexpensive 'shop lights' - 4 foot florescents with 2 bulbs. They can be purchased at various stores for anything from 8-10 dollars or on sale for even less.

You'll also need some sort of sterile soil-less mix for germinating and to prevent the damp-off and a small fan is also a great help. As to starting containers, you can use just about anything and there is a long thread just a bit down the page full of suggestions for containers to use. Once the seeds germinate and develop their first set of true leaves they have to be separated, transplanted to individual containers, and fed regularly with a mild 1/2 strength fertilizer solution.

As an alternative, investigate the Winter Sowing forum for an alternative method of growing from seed and doing it outside. You may find their approach appeals to you more.

As to when to start - the usual recommendation is 6-8 weeks prior to your last frost date - info readily available from your local county extension or online. As already mentioned, gourds are usually direct seeded but in your short growing season you may wish to start them early and transplant instead. Broccoli is a cole crop and prefers cooler weather and can even tolerate frosts so it goes out much earlier than any of the others you list.

Check out all the FAQ's that exist (linked below) here for all the basic info and good luck with your garden. ;)

Dave

Here is a link that might be useful: Growing from Seed FAQ's