The silk itself is an incredible chemical composition that may be ejected through 16 or more openings at the end of the spider's abdomen, manipulated and wrapped to increase its strength. As the spider builds the web it places a series of spots of "glue" on certain strands, and it is this glue that is the actual stickiness of the web, not the silk itself. The spider that builds the web then instinctively avoids, as much as possible, those strands that have the glue on them, choosing instead non-glued strands when they need to race across that web to grab, bite, and paralyze some insect that has blundered into the web. But, as added insurance the spider also exudes some oil onto the tips of its feet, and it is this oil that further prevents it from getting stuck on its own web should it come into contact with that glue. So, without the glue there is a chance that a spider could be stuck on its own webbing. 

With this in mind I suppose the answer to the second question regarding other spiders' webs would be that any spider whose feet are properly oiled could walk on the silk of another spider and not get stuck in the webbing. However, perhaps a spider on the wrong web might have a greater chance of being stuck if it did not properly recognize the strands that were glued versus non-glued, whereas on its own web it probably feels right at home and more rapidly "knows" which strands to use for travel. And, spiders probably are likely to avoid the webs of other spiders, particularly webs of other KINDS of spiders, as the structure of the webs of different kinds can be very different.