Ya gotta give it some bark!
Close, but I think another answer fits better.
Motels are very stationary!
Eleanor or Montgomery Lift?
- Angels always exclaim when they see me climb inside a horse carcass.
Lo, DiCaprio.
- I’m what you’d do to the fourth letter of the alphabet after you marinade it but before you smoke it.
(Thanks for the additional clue!)
Rub Dee.
(And now that I have thought about it, thanks for making the original clue about barbecue…)
Are you hungry? 
These work, but mine works better.
Not for barbecue… 
After IMMENSE PRESSURE (the disadvantage of living with the primary clue-maker), I have arrived at the solution to this. Roy Blunt, Jr.
Sheesh. 
Donald Rump?
You got it! Yayyy!
(If it were a cryptic crossword clue, I would add “& lit”)
- When I have a fever, my brain tends to get scrambled.
Eggy Lee. (I was spending all my concentration on “brain” and not on “fever”!)
Edit Head?
36. When it comes to diamonds, I always blow out the competition.
Rock HOLT! (Diamonds, see?)
(I have a feeling I am not right.)
Yayy!!! Another correct one.
Sadly, no, but I actually like that answer. Bonus points!
- Yes, yes, yes, I think it’s so nice to have you back where you belong.
“Hell, Dolly, I’ve told you that a million times!”
- I’m a diminutive actress who would feel right at home in a grass house.
Tipi Hendren? IMDB says she is 5’4", and google says “tipi” is an alternate spelling of tepee/teepee.
- I’m still dead, but I’m worth about $1.02 now.
Buck Owes?
No, TH is much taller. But her last name does start with an H.
Ha! But no.
- I’m a diminutive actress who would feel right at home in a grass house.
Linda Hun? I don’t know where the Huns lived (and google is no help, keeps wanting me to look at “hen houses” or “HUD homes”) but it is possible they lived in grass homes.