I suppose you can store them in the fridge until September.

Many bulbs growers and gardening books suggest that these bulbs be planted after the ground has cooled.

But I have been wondering why then bulbs in the garden don't need to be lifted after blooming.

Daffodils, crocus and other bulbs in my garden have been blooming year after year after year even though they are left in the ground all summer and winter. Whenever I unearth any daffodils or other bulbs while gardening, I simply dig a new hole and replant those bulbs right then and there. Why then can't we plant new bulbs in summer and leave them be as long as we don't saturate the ground and causing the bulbs to rot?

I guess, my concern is the temperature in the fridge (too low perhaps?). If you dig up bulbs after they bloom, it is recommended to store them at 60-65F in a cool basement (I don't have one) till fall planting. My concern is - are they going to start growing/blooming when I take them out of the fridge and put in the soil? What if the soil temperature higher than temp in the fridge on Sept 30th? I suppose, I need to find a thermometer to measure how cold fridge storage is.

I wouldn't store them in the refrigerator until you plant. Just keep them in a dry, shaded area in their mesh bags. Do not store them in plastic.

According to the MN Daffodil Society website, you can start to plant daffodil bulbs now.

See their daffodil schedule at the link below.

The times for planting tulips and crocus are similar, and it should be fine to also start planting them now.

Here is a link that might be useful: Care activities by month

Yes, as someone pointed out...the garden fairy doesn't come and lift the bulbs every late spring and leave them in her shed till fall planting time. They stay in the ground..that's where they belong.

Bulbs sold in stores are usually 'cured' for shipment, that means they've been dried for a few weeks to lose almost a quarter of their weight. If you've ever grown onions it's the same curing process, you can't store them wet since they'll 'sweat' out the moisture and end up moldy and rotting in storage from the dampness.
Because they've been cured, they store and ship well but are in a deeper dormancy than bulbs you have in the soil already. If I dig up my daffs now they might already be sprouting roots, but the ones in the garage won't be planted for another couple weeks since the warm damp soil may cause them to rot before they start waking up.
Uh so what Chemocurl said, store dry and plant when it's cooler.... No refrigeration necessary.

I have measured temp in the fridge - its 46F, a bit too cool for recommended storage conditions. Now I have unloaded bulbs into shallow cardboard boxes and put them under the cooling vent in my bedroom, we have no cool basement in this house (elderly relatives turn on gas fireplace downstairs because 72F cooling is too cold for them). I am still prepping new flower bed for the bulbs, I think planting daffodils now is a bit early - it is currently warm, humid, and raining every two days or so in MN.
Thank you, everybody for your suggestions!