Alt tags are used to convey information to people who cannot see the images, so they should always be used (unless the image is just decorative). If the images are different, use different alt tags.

Yes, there is a down side to putting them in both places and that is duplicate content. You can use cross-domain canonicals to indicate which is the original one, but if you don't have control over the other site, you cannot be sure they will cooperate.

It can sometimes make sense to submit content to a site that has higher visibility and traffic than yours to get the exposure, but in general the best place to put your own content is on your own site, certainly if you want to rank for target keywords.

Also, you mention that you had some articles created for a client. Keep in mind that if this is content that is churned out in bulk, it is not likely to do well. You should make a point of using unique and high quality material.

Wait till Google crawls your page again. You will know when this is by checking the date on the cached version of the page, which you can find by doing a search for the URL of the page you are interested in, then clicking on the little green down arrow next to the URL in the search result.

That is not the picture I see (I see a street view) but if you go into your Google places account (I am assuming this is your site) you can upload pictures and make other changes.