There are countless Villa Rental sites in Italy and many travelers have used one or many of the other sites. What really matters is if you find one, whether it be on Homestays or not (they are reputable) that meets all of your criteria and at a good rate. No it will not be too cold in Italy in June, it will be quite warm, in fact, especially in Tuscany and in Lake COMO. You might get lucky and find a pool that is heated but otherwise the sun is so warm each day, that I think the water will be quite warm enough to swim in. I've been to the Lake District in June and to Tuscany in May and indeed it was quite warm and beautiful. Note you should book as early as possible because these two areas are among the most popular for American tourists and while there are many, many villas, you are going to want to get first dibbs on your tops choices in the areas you are most interested in.
Where will this matter most? To your husband who wishes to cycle. If he has not already, I would suggest he google this year's Giro d'Italia race, which was not too long before the TdF and see what the race route was so you can possibly set up housekeeping not too terribly far from where some of the mountain stages might have been. For me, this would logically be more in the Como region as you are once again quite near the Alps and the Dolomites as well, so it's quite likely that in Como your nearer to the climbs. I don't know that the Giro is as known for its climbs like the TdF is, otherwise many riders would not compete in both races in the same year as it would be too physically demanding. That said, there are surely some Class 1 or 2 climbs in the Giro but doubtful of any HC climbs in that race.
OK here's the link I found the map for the 2010 race (not as easy as I thought it would be!)
So as I suspected, the climbs are in the Dolomites, not the Alps and the race route info that I read on other sites stated that the hardest part of the race was the end, where the mountain stages were and voila, that's where they are. So Como would be the closest to that area, though it'd still be a drive to get to certain points on the route - I assume that for a month you'd be renting a car? So your hubby can drive his bike over to the routes he wants to take and go from there. The race concluded in Verona, which is flat. There is a part of the race that went through Tuscany as it went to Montalcino where they also happen to make some very fine wine! Tuscany is hilly, but by no means mountainous.
So I don't know what your make/break options are, but everyone wants to have a good time. I imagine now if you start looking for a nice villa in Como, you'd have the TIME OF YOUR LIFE. The kids would be happy with the lake and the pool to amuse themselves with, the little towns along the lake, Oh so beautiful setting - it's pure heaven up there, I kid you not. Plus hubby gets to put on his Giro helmet and take the rides. As for more rental companies, I can find you more easily as there are many that several of our members here have used before with great results. Most of the companies over there enjoy great reputations because they rely on US to come and keep them busy!!
Let us know if you need more help and information!