Short answer: no, you do not need to pay the apprenticeship levy on a GBP 180,000 annual pay bill. Your understanding of the GBP 3 million threshold is correct, and here is how it works so you can be confident about it.
The threshold
You pay the apprenticeship levy at 0.5% of your annual pay bill only if your annual pay bill is more than GBP 3 million. There is a GBP 15,000 annual allowance that reduces what levy-payers owe, but it only matters once you are over the threshold. With a pay bill of GBP 180,000, you are well under it and owe nothing.
What counts in the pay bill
It is the total pay bill, not just basic wages. HMRC defines the pay bill as all payments to employees that are subject to employer Class 1 secondary National Insurance contributions, which includes wages, salaries, bonuses and commissions. So yes, you were right to ask: bonuses count. In your case, even including everything, you are comfortably below GBP 3 million.
One caveat: if your company is connected to any other companies or charities for Employment Allowance purposes, their pay bills are combined with yours and you get just one GBP 15,000 allowance to share between you. If you are a standalone company, this does not apply.
Is there anything to report if you are under the threshold?
No. You do not need to report the apprenticeship levy on your Employer Payment Summary (EPS) if you have no levy to pay. There is nothing to declare to HMRC. Just run your normal PAYE as usual. HMRC's guidance is clear that reporting only starts when you have an annual pay bill over GBP 3 million (or expect to exceed it in the tax year).
Is the levy connected to employing apprentices?
No, the two are entirely separate. The levy is a payroll-based charge paid regardless of whether you employ any apprentices. It funds apprenticeship training via the government's apprenticeship service. Taking on an apprentice next year is a separate decision and does not trigger the levy, nor does paying the levy entitle you to anything special. Even as a non-levy payer, if you take on an apprentice you can access government funding towards their training costs.
Practical note: the threshold applies each tax year, so keep an eye on it as you grow. But at GBP 180,000, you can safely ignore the apprenticeship levy for now.