Part of the problem is that the standard conception of an adventurer is somewhat unrealistic in and of itself. An independent group of characters who possesses a great deal of material wealth and specialized equipment yet travels so widely that they seldom cross the same patch of ground twice is extremely rare in realistic settings.
With that said, there are a few things you can do to make it more workable:
Be prepared
The biggest thing that can help you out here is to get information ahead of time about the adventure you're going on. Figure out what you need, and what you can leave behind.
Strip down to the essentials
Look at Strider in the Lord of the Rings movies as an example. His gear is extremely streamlined -- He carries no tent, no utensils, few rations, little in the way of personal effects. He uses his survival skills and a bare minimum of equipment (some means of starting fires, a cloak in place of a blanket, and so on).
Naturally, you'll need some form of survival skill to make this work. And the specific environment you're in will determine just how much you can throw away.
Quick release packs
For small loads, you can make it a practice to drop your packs when combat begins, and then pick them up again when the immediate danger has passed. This still leaves you somewhat vulnerable to surprise (and traps) but otherwise allow you to function "above load."
Caches
Bury your excess stuff at critical locations. If you're in the process of bringing a haul back to town, bury some of it at the dungeon and make several trips. If you're about to leave town for the adventure, bury any gear/food/etc. you won't need. Make a habit of creating survival packs and hiding them in areas you frequent.
This was a tactic employed by real-world people who had more stuff than they could carry (I most recently read it in relation to people taking loads of gear up into gold country during the northern gold rush).
Use magic
Bags of holding were invented for a reason.
5 weapons is correct if you always choose the option with the most weapons. Though this probably isn't the best use of your initial equipment loadout.
You could list them all on your character sheet, but it might be better to note your attack bonus (your strength mod plus your proficiency bonus plus any other bonuses), and your damage modifier (your strength plus any relevant bonuses), and then either write down or remember the die that you roll depending on which weapon you're using.
This is probably easier than writing the same expression down several times with a different die in each one.
There are only limits on what kind of attack you can take with a specific weapon if specified. One such limitation is for two weapon fighting which requires light weapons.
Best Answer
Your starting equipment is your equipment you start with at first level. Gaining a level in another class adds no additional equipment.
Player's Handbook errata clarifies this:
Another question might be that a level 10 character should have more advanced equipment than a starting one, which is intended to be used by level 1 characters. That's covered in the question What's the starting wealth for higher levels?