I have a lot of old weathered, grey non-sticky railroad ties. Is there a way I can use them to build raised beds? Maybe by lining them with cedar?
Yes, you can use them. Most people line them with plastic to keep any chemicals out of the bed that could be harmful, but it sounds like, if they are grey, the creosote has washed away and most of the chemicals have already been leached away.
Would a Styrofoam ice chest make a good container for planting vegetables?
As long as you provide this container with adequate drainage holes, it should work just fine. The following article should be of some help to you:
I have a question. Can I use the new treated lumber to build my raised beds?
The pressure treated wood sold in today's stores are required to be non-toxic, so are therefore considered safe for gardening. But some people still feel unsure about them. If this is the case for you, you can also look at building out of cedar. While it is more expensive, it will last longer.
The following article should be of some help to you: https://www.gardeningknowhow.com/edible/vegetables/vgen/raised-vegetable-gardens.htm
Could I put a layer of wood ashes, a layer of shredded newspapers, and a layer of dried lawn grass from last year with a layer of dirt between each in a raised garden that I want to make?
Yes, this is fine and will not only break down over time but will fertilize and nourish both the soil and plants. This article will help you with that: https://www.gardeningknowhow.com/special/urban/raised-beds-for-urban-settings-no-digging-required.htm
Can I put the soil for a raised vegetable bed straight onto the lawn, or do I need to dig the turf out out, or cover it?
These articles should help: https://www.gardeningknowhow.com/edible/vegetables/vgen/raised-vegetable-gardens.htm
Staying away from pressure-treated lumber, how do I protect the insides of my pine deck boxes that will be used for square-foot vegetable gardening? Built about eight years ago, it's necessary to build new ones due to rotting from the inside. I'd like the new boxes to last longer.
The pressure treated wood sold in today's stores are required to be non-toxic, so are therefore considered safe for gardening. But some people still feel unsure about them. If this is the case for you, you can also look at building out of cedar. While it is more expensive, it will last longer. Another option is to line the beds with plastic. I have used the heaviest plastic drop cloths they have in the hardware store for this, and others have suggested rubberized liners for roofs and manure troughs that are available through feed stores.
I let my garden go to weed last year (laziness and too much work away from home). Is there something I can put in the soil of my raised bed garden to kill off the mass of weeds and still be able to put in vegetables this year?
I would just till the bed under to get rid of the weeds. It is fastest. If you do not have a tiller, you can rent one from the hardware store or check places like CraigsList.org for people who will till your garden for a small fee. If you ask around in your community, you may even find someone who will till it for free.