So as for the tree diagram, realise that there will be more than one diagram,in fact one diagram starting at each pet as any could be chosen first and all the remaining pets could be chosen second.
A list might be easier, abbreviate the animals to the first letter of their name for instance:
Showing CS=Chita first, smudge second and so on.
As for the probability that one dog and one cat, realise this condition can be met in two ways:
A cat picked first then a dog
Dog first, cat second.
So you need to add the probability of those two scenarios together as its one or the other.
When working out the probability of each scenario, realise there are two events happening one after another so you must multiply the probability.
so for the first one:
cat picked first (3 cats out of 5 animals) so 3/5 chance of picking a cat.
Dog second (2 dogs still in waiting room out of now 4 animals left) so 2/4 chance
therefore multiply 3/5 by 2/4 = 6/20
You will need to add this to the dog first cat second probability.
As for all maths, work out a rough answer in your head first which you can come back to compare with later. for this situation you know there are roughly speaking equal numbers of dogs and cats so the chance of picking one then the other isn't slim so if you get an answer of 1 in ten million something went wrong, the same if you got a 99/100 scenario.
As for the second question it is about compounding its value (similar to compound interest). I.E it goes up on 8% of its value each year, starting with a new value each year. imagine if it was worth $100 at the start, it is worth $108 at the end of the first year, but isn't worth $116 at the end of the second year as 8% of 108 isn't 8, its more than 8.
For this example as you are not given the initial cost and it is all in percentages (8% and double value) you can take any initial value you want, so to make an easy start, pretend it cost 100 initially.
work out what its worth after each year at a time.