I’ll answer point 2 for you.
Twitter will tell you thta the answer is no, you should not include the tm Twitter or Tweet in your application name and if they find this they will shut you down.
As Twitter is a closed ecosystem i’d advise you to follow their rules however… however…
If you could care less about twitter and have no reliance on their platform then i direct you to the following…
With regards to trademarks combined with other words in a url there have been a number of legal changes lately about the use of trademarks in domain names. This issue was litigated recently in Federal court (9th circuit) - Toyota Motor Sales, U.S.A., Inc. v. Tabari, 610 F.3d 1171 (9th Cir. 2010). (It is our understanding that this is the most current legal case on this issue) where the 9th Circuit noted that "the nominative fair use doctrine allows such a truthful use of a mark, even if the speaker fails to expressly disavow association with the trademark holder, so long as it’s unlikely to cause confusion as to sponsorship or endorsement. See [Playboy Enters., Inc. v.] Welles, 279 F.3d [796, 803,n.26].
This also doesn’t prevent anyone from registering the domain www.TwitterSucks.com or similar which has been a long standing position based on the Taubman Willow bend case 319 F.3d 770 (6th Cir. 2003)
But hey i’m not a lawyer and i dont even play one on TV so seek your own legal advise first. Just understand that Twitter WILL close down your app when they find it.