1) can I glue the slats together and make suitable strong table legs and other structural parts, 80x80x900 e.g.
Yes. This is quite commonly done.
2) if I then cut mortise/tenons in there, will they then be strong enough, or will the wood break up?
Yes this is fine, particularly if you orient the glued-up legs so that you're cutting the mortises through the continuous face that will be on two sides of each leg. However, this construction affords another option and that's to form the mortises during the assembly process by leaving the space in one layer of the lamination, see Simplifying construction in the last link at bottom.
3) for the bench top I have considered either joining the slats horizontally with two layers to produce a 40mm thick top, or putting the slats on their side to produce an 80mm top.
Either of these could work fine in practice. The second option would generally be considered the better of the two by woodworkers.
The second option will make for a very heavy, stout benchtop (often desirable, especially for a bench intended for hand-tool work exclusively or primarily) but bear in mind the eventual weight of a beech slab 80mm thick! This method of construction has another advantage in many cases, with most boards being flat-sawn or plain-sawn turning them on their sides and glueing them together face to face makes for a top that is effectively quarter-sawn wood. This orients the main seasonal movement direction up/down and not across the width which comes with some advantages including, in general, greater stability.
4) if I can't find long enough pieces to make the table top, and instead make the top up using overlapping shorter pieces, will that make a strong enough table top?
Yes. This is commonly done in the manufacture of modern built-up work surfaces and benchtops.
The staggered joints, although individually weak, become essentially irrelevant because they're supported by the wood around them. You want to try to space them out sensibly for maximum strength, three close together could be a weak point, but the entire structure becomes so strong you don't have to be overly concerned.
You're going to need a lot of clamps to do this properly, and for best results they should be strong because you want to achieve high clamp pressure. The number of clamps becomes more important the thicker the piece being glued because clamp force spreads out from the point of clamping.
Clamps cannot be too tight and the tighter you make them the better and more reliable your joints will be.
Some further information from previous Answers will be of great benefit to you as an inexperienced woodworker:
How long after cutting wood can I wait to glue?
What do I need to do to prepare wood for gluing?
Should I orient the cupped sides toward each other when laminating boards?
How to join short boards to make a longer panel
What is the benefit to gluing boards together instead of buying a thicker board?