If I restart Fable 3, do I get to keep any of my weapons or equipment from the previous game?

Since there doesn't seem to be a way to try alterante endings (without restarting and losing my save), do I at least get the benefit from previous run throughs?

No, your progress on previous games does not give you any direct benefits in a new game. Items you purchased (or obtained for free) from the Sanctuary Shop, like the Yule Costume, are available across all saves, and some statistics are tracked across all your games, and of course your knowledge of quests and areas and the results of your previous decisions will help you in new games, but when you create a new hero, you'll be choosing between a basic sword and a basic hammer and between a basic rifle and a basic pistol.

You should find a trustworthy friend to give him the items you want to keep. Then, have him give you back the items with your new character.

No. Each save-file is radically unique.

Create a template primary Key-master, who is your "serious" Hero, and then create several secondary local, non-"Gold"/$ differentiated Gamertag identities or "sub-Heros" in order to be able to effectively conserve and trade what you desire. No trustworthy friend (unlikely in modernity) needed.

Also distinct (local, non-monetary) Gamertag profiles allow collection of all Relic weapons etc. The Key-master can immediately jump into any new Hero game in the Bowerstone Castle garden area and unlock the Relic.