Demand for goods:
\(\displaystyle Z = a_0 + a_1Y -a_2i\)

Demand for money:
\(\displaystyle (M/P)^d = b_1i - b_2Y\)

1. Find the equilibrium level of output
The question tells you exactly the steps to follow so im not sure where you are stuck. Start by finding the IS Curve, which gives the combinations of Output and interest at which the goods market is in equilibrium.

Goods market equilibrium: Goods Demanded = National Income
\(\displaystyle Z = Y\)

\(\displaystyle a_0 + a_1Y -a_2i = Y\)

\(\displaystyle Y(1-a_1) = a_0 -a_2i\)

\(\displaystyle Y = \frac{a_0 -a_2i}{1-a_1}\)

This is the IS Curve.

Now, find the LM curve. This gives all combinations of interest and income at which the money markets are in equilibrium.
Money market equilibrium:

To find equilibrium, find the point where the LM and IS curves intersect
IS: \(\displaystyle \displaystyle Y = \frac{a_0 -a_2i}{1-a_1} \Leftrightarrow i = \frac{a_0 - (1-a_1)Y}{a_2}\)

So,
\(\displaystyle \displaystyle \frac{b_2Y + (M/P)}{b_1} = \frac{a_0 - (1-a_1)Y}{a_2}\)
Solve for Y.

2. Suppose the government increases spending by 1. Which parameter of the model changes
I cant explain this to you. You need to learn the definition of the model. Read your notes and see which equation you think normally has government spending in it.

Government spending is part of the demand for goods and services, so it appears in the goods market equation. It is assumed to be exogenous (constant) in basic ISLM models. So the anser is a0.

3. how much does equilibrium output increase in the above example?
Look at the equation you obtained for the equilibrium output level in part 1. How much will output increase if you increase the parameter by 1?

4. Why does output increase by less than the IS Multiplier
This is called crowding out. Government spending increases activity in the goods market which increases the demand for money. This leads to an increase in the interest rate in the money market, which puts downwards pressure on investment. The suppresed investment reduces income slightly.