Feb. in Europe on the cheap

Assuming the budget doesn't need to include airfare and you avoid cities during major local festivals, I think you could just about do this in southern Spain (generally warmer) or Sicily.

Europe is having strange weather this winter, with snow and cold in places such as Greece that don't often get it. February is usually the coldest and wettest weather in Europe. To get the temperatures that you are seeking, you will need to head to head to southern Spain or Portugal. You can easily find a monthly rental apartment for €1,000 a month. It will be more expensive if moving around and taking weekly rentals.

Europeans seeking those temperatures would also consider Morocco, the Canary Islands, Madeira or Cape Verde.

For warm, go south. For cheap among the warm options, I'd go with Spain. Plan Andalusia.

You can plan for a vacation in Europe in 2018 but it may be hard to budget that far in advance, since who knows what the exchange rate will be then. I would agree for 60 degree weather in Feb, southern Spain, Portugal, Sicily and maybe the far south of Italy would be your choices.

Greece can also be quite cheap, especially in the off-season. $170/day for a couple can stretch pretty well.

Many of the countries mentioned above (but obviously not Morocco or Turkey) would also have Mardi Gras/Carnival events mid-month. https://www.ricksteves.com/watch-read-listen/video/tv-show/european-easter

You've got 3 numbers in your question. The first is irrelevant (presuming you're in good health). The second is pretty easy. The third is just not likely, mostly 50s is probable, but over a month, you're likely to get some days in the 40s and a few in the 60s. Though it's worth noting that this year is uncommonly cold and wet in Italy, but Spain seems to be avoiding it.

Andalucia is probably your best best - it is the warmest, driest part of Europe. Except for about a week in Barcelona (there's some annual convention or trade show that spikes hotel prices), Spain is low season. Barcelona's weather is generally mild in winter and even Madrid can be pleasantly balmy. You could find decent hotels for well under €100/night. That leaves you plenty for food, attractions, trains and even a rental car here and there. Anyone over 60 can get the "gold card" (tarjeta dorada) for discounted train travel and most sights give a reduction to anyone over 65.

Italy's probably more expensive (though you'll still find lots of hotel bargains) and both chillier and wetter. Sicily may be even cheaper than Spain, but colder. Most of the places to visit there are on or near a coast, while Andalucia is mostly inland.

With hotels at those rates, I would expect apartment rentals to be equally attractive.

The Algarve, Portugal's south coast, has traditionally been a winter refuge for chilled residents of the UK, much like snowbirds in eastern North America flock to Florida. It's worth remembering that the ocean is the Atlantic rather than Mediterranean so only the hardiest could dip in a toe. In Canada it is easy to find package deals to the Algarve, at least from cities where Portuguese have settled.