My yahoo has been messed up and I haven't been able to reply for  hours!  making me crazy!  ::smile::

ASL grammar is not as simple as merely taking out all the small 'function' words in the English and you're good to go...
My translations of your sentences:
1.  CAT ix WHITE  (ix refers to indexing, basically pointing.  If you don't index, you are saying 'A cat is white', not 'THE cat is white'. There would be a head nod accompanying this.
2. this sentence would vary highly with where "there" is...  is someone on the phone? at the door? in the office next door?...  each one would be signed slightly differently.  WHO THERE with wh question marker (eyebrows lowered, head tilt forward, last sign held) will work, so will  ix WHO (also with wh question marker) and so would WHO ix WHO-- ASL often repeats the wh question word at the end of a question, and it is the sign that would be held.
3.  YOU LOVE ME works, it would have the y/n question marker-- raised eyebrows, head tilt forward. 
4. As someone else mentioned, this one varies with where the speaker is.
5.  TODAY WEATHER WHAT (rh-q) SUNSHINE is how I would probably sign it.  Its hard to tell with sentences in isolation, but the rhetorical question adds an element of ...  I dunno... suspense? that seems needed to me...
6.  YOU NAME WHAT, NAME YOU, WHAT NAME YOU WHAT, true ASL signers do not often use YOUR, even though that is how it is asked in English.  It would more accurately translate as 'what are you called' rather than what is your name.  All require the wh question marker
7.  EYES MY BROWN with head nod.
8.  DICTIONARY MY WHERE with wh question marker.
For both 7 and 8, ASL tends to follow a topic/comment structure.  WHERE MY DICTIONARY is not wrong, per se, but since the question is about the eyes, or the dictionary, they would go first. And ASL tends to put the question word at the end, or at the beginning AND end, only rarely just at the beginning. This type of sentence has a topic marker over the topic, then they would change to the wh question marker over the question word.
9.  YOU YOUNG YOU.  ASL frequently repeats the pronoun in  'predicate adjective' sentences like this one.  Change the vocal intonation of the sentence just slightly though, and it would change the translation.  YOURSELF YOUNG WOW if there was emphasis on the young part. 
10.  KID ix PLAY++     KID is your topic, the ix makes it THE kid, PLAY is repeated to make it present progressive tense.  topic marker over KID ix

Really, if you are not fluent in the language, I would highly recommend getting signing models that ARE fluent.  You will get better language in your demo that way.  Signing in English word order has its place in the world, but if you are aiming for ASL, get someone who knows ASL to model for you.  I would be happy to write these sentences in the signpuddle for you if you like...