Lightroom :: Color Space When Exporting From RAW

when exporting a photo, does Lightroom convert to a profile or assign a profile? Because there is no way to choose. I tried to export a photo with 3 different color spaces (sRGB, AdobeRGB and ICC profile from laboratory where I print my photos). After exporting them to JPEG it turned out that all of them look differently on my monitor - does it mean that Lightroom assigns a profile? If it was converting, shouldn't they have the same colours? What is more, after printing them in laboratory, results were completely different than I expected - the photo which had closest colours to what I saw in Lightroom was that in sRGB, but that with ICC of Lab was very different (much colder colours).
 
Where is the problem, or what aspect do I seem to misunderstand? Do I have wrong settings, should I use DNG to work with photos, should I export to TIFF, or I just have too weak monitor or wrongly calibrated one? Should I calibrate when viewing a picture in Lightroom or with the use of a photo exported to the ICC profile of Lab?
 
I would like to have a little bit of control over what I'm working on, depending on whether I want to publish it on a website or print. I know that my monitor can be a problem (I have an iiyama with IPS), but surely there has to be any way to make results of my work a bit closer to my expectations.
 
Just for information, my workflow doesn't require Photoshop, as I rather prefer to use only tools from Lightroom. I hope that my problem doesn't require the use of Photoshop.