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You may not move any pieces until you have placed your Queen Bee; the Queen can be placed any time from your 2nd to your 4th turn and must be placed on your 4th if not placed earlier (p.1).

Except for the first piece each player places, a newly introduced piece may not be placed touching any of your opponent’s pieces (p.1).

No — the One Hive Rule requires all pieces in play to remain linked at all times, so you may not move a piece that would separate the Hive (p.2).

Most pieces move by sliding and cannot move through a narrow gate or into a space they cannot physically slide into; if a piece is blocked so it cannot physically slide out, it may not be moved (exceptions and details on Grasshopper, Ladybug, Beetle noted) (p.2).

A Beetle cannot be placed directly on top of the Hive when introduced, but it can move on top later; while a Beetle is on top the stack takes on the Beetle’s colour for the purposes of the placing rule (p.1).

Once in play the Mosquito takes on the movement characteristics of any creature it touches at the time (including opponents'), but if it is touching only other Mosquitoes (including stacked Mosquitoes) it may not move; if moved as a Beetle on top of the Hive it continues to move as a Beetle until it climbs down (p.2).