I'll go through your questions one at a time:
Unfortunately you can't place more than one URL within a twitter account, twitter is intended to represent a single entity (that's why). I would advise you to perhaps have a separate twitter account for each website you wish to represent using the service, that's the recommended practice.
You can't, Twitter is intended to be simple, if you want to list your phone number, have it on your website. The idea is that people will enjoy your tweets and therefore visit your site for more information and details of how to contact you. It's a messaging system, not a forum profile or telephone directory.
You could put your logo in the background as part of a background image (in fact you could have a text background which has stuff like all your URL's, your phone number and anything else you want to list) - granted people won't be able to click on it, but stuff like a phone number just requires being visible, not being clickable.
I don't think it makes it look terrible, but it can look bland if there's little personalisation.
Sounds to me like they might have followed you, sent the DM and then unfollowed you after.
There isn't a way to get around that, but it's a rather flawed way of looking at it anyway, when people search for your company their looking for your website, not your social network profiles. You should list your accounts on your site to help people stay in touch and you should link to your site from twitter (so new followers can find you), but thinking that people come looking for your business to see what you're tweeting doesn't make it seem like your business has a lot of value.