Three problems below with my attempt solutions:
1) How many ways (ordered selections) can the letters of the word HULLABALOO be arranged?
$$\frac{10!}{3!2!2!}$$
2) How many distinguishable arrangements of the letters of the word HULLABALOO begin with the letter U and end with the letter L?
I took these to be accepted positions, and thus the letters are reduced to HLLABAOO
$$\frac{8!}{2!2!2!}$$
3) How many distinguishable arrangements of the letters of HULLABALOO contain the two letters HU next to eachother in that give order?
I joined HU into a single letter $\alpha$ and now I have the letters $\alpha$LLABALOO:
$$\frac{9!}{3!2!2!}$$
Are these correct?
Yes, these are all correct - standard approaches to solving these types of problems.