At this point, no one really knows what the weather will be like in June of 2016. I recommend checking out
HydroGuy's Detailed Weather Data for DLR thread to get a sense of the historic weather in June, including the historic maximum and minimum temperatures. I generally agree that the weather should still be okay in June, but you never know...you might get there when an unexpected heat wave rolls through Southern California (it has happened).
In terms of crowds, I would guess that the park is going to be busy, but it shouldn't be capacity-level busy. It is the last Monday before start of the SoCal Select Summer AP Blockout and some schools are out (although, two fo the biggest in Southern California (LAUSD and San Diego) are still in session). There isn't a grad nite on the 6th (there is one on the 7th), so there should be extra guests in the parks for that. Overall, I haven't heard of anything that will increase the crowds above what would normally be expected for a Monday in early-June, although, as noted earlier, crowds have been generally higher than historic since the start of the 60th Anniversary.
Attraction-wise, there should be to many things that are closed, other than those that are part of the year-long or more closure (Mark Twain/Columbia, Tom Sawyers Island, Canoes, and the train). As mentioned, F! will be dark and it's not yet known if the new Frozen stage show at the Hyperion will be open. That being said, things do close unexpected (or with very short notice), but we won't know what is scheduled to be closed until 45-days before hand or less.