What are IRA contribution rules?

Monthly contributions are not required by law.  You can contribute any way you like as long as you don't exceed your annual limit.

So...you can suspend contributions and pick them up again or you can make a single lump sum payment to catch you up.

You can make contributions any way you wish provided you don't contribute more than the annual maximum for a given year.
You can do one lump payment, monthly, quarterly, every other month, really whatever you want to do.
You also have until you file your taxes for a year to make a contribution for that year. For example, you will have until April 15, 2016 to make a contribution to your IRA for 2015. So, you could conceivably do one contribution per year even up to about 3.5 months after the close of the calendar year.

Statistically, assuming you're investing this in primarily equity indices, the market goes up more than it goes down. That being said, you would have been historically better off "plunging" the full amount in than "wading" in each month.

Statistics don't drive most investors' decision making though. If you are uncomfortable with the current state of the market or are digging deep into your pockets for the annual contribution, it likely makes sense to contribute monthly. If this is the route you take, make sure your contribution is automatically invested or that you go in and proactively purchase the anticipated positions. 

Actually, the best time to make an IRA contribution is the first business day oj January in the year you are making the contribution for. That wya, yul'll have the entire year to grow the money tax deferred.

There is no requirement to make a monthly contribution. Monthly is one suggestion if you cannot afford a lump sum contribution simlpy for the discipline it creates for saving.