Build a regular expression to define a regular language
One way to do it by hand, in such a simple case, would be to build a FA for
condition 2, i.e. strings that start with $1a$, and another FA for
condition 3, i.e. strings that contain $aa$.  Both FAs have only 3
states.

Then you use the intersection construction to have an automaton that
obeys both conditions. That remains simple enough if you organize your
state pairs as a 3 by 3 matrix.

From that you can easily deduce a regular expression:
 $1(a+a(a+b+1)^*a)a(a+b+1)^*$

Your solution can be proved consistent (compliant with the stated conditions) with a case analysis over the two main star operators (cases depending on whether there is zero occurrence or at least one). One also has to prove it is complete, i.e. that it does cover all compliant strings.

If you have a constraint regarding the use of parentheses, you should state it in your question. Is there a reason for the constraint ?

Find more details in the answer to your other question about this language, and a comment on how to better write the question.