Looking good so far!

Regarding your questions, I may have a few ideas:
1. Considering the backstory, how about names such as; 'Avenger', 'Vengeful' or 'Martyr of Gryphonne' (Just from the top of my head

) Generic names such as 'Thunderer', 'Leviathan', 'Behemoth' etc. also work quite well. May I suggest taking a look into medieval English (or European for that matter) history - they often named their siege engines with a variety of creative names. For example 'God's Sling' was applied to engines such as a trebuchet - how about 'The Emperor's Sling' or some other name. Just some thoughts to get you going
2. If I recall correctly, the Space Marine Vindicator has a crane included in the kit for the exact same purpose - check eBay for bits regarding that. If it's not the aesthetic you were going for, then how about finding some historical engineer vehicle 1/35 scale kits and nicking the crane off them. For example, this 1/35 Challenger ARRV:
http://www.hlj.com/product/ACMC024
3. I think putting in a counter weight would be a bad idea myself. Simply because, unless you have it finely balanced, it'll either slump forwards or it will slump back to it's full extreme. Personally, I'd suggest possibly using some magnets in the gun mount. If they are strong enough, then the gun will be prevented from tipping forward or backwards too far by the strength of the magnets holding it in one position.
4. I've been waiting for someone to ask this for ages!

If you want to go down the route of buying ammunition bits, there is myriad places you can get them - anvil-industries, Forgeworld, shells out of a Vindicator etc.
However, if you want a low cost solution, then I've found that casings from a .22 Rifle round are the perfect size to represent shells. All you'll need are a handful of
spent .22 casings (The empty casings ejected after the round has been fired) and I'm sure you could either pop round to your local game hunter/rifle enthusiast/casual shooter/shooting range, and ask for a handful of spent rifle rounds. (Obviously it's important to remember that you'll need to explain
exactly why you want them and depending on your age, the law and how dangerous the shooter views spent casings, they may ask for
ID or a Firearms license first). Without the heads, they are perfect for throwing around the gun deck as spent rounds or stacked in a corner to represent the tubes of propellant used in modern artillery.
Of course, to represent live shells is a touch more difficult if you want to use .22 rounds. Obviously, for legal and safety reasons - live ammunition should NEVER be used in any circumstance. Even if it looks right -
it is downright 100% dangerous and irresponsible to venture outside a designated shooting zone with live ammunition - Duh! (I just feel the need to reinforce that point

)
Therefore, to represent 'live' shells with a .22 casing, I would grab a spent casing and then mold a little
GS head in lieu of a real one. The result looks very realistic and is very low cost. I would recommend using a metal primer to prime the casings - I have attempted it with a simple matt black primer, but it didn't etch into the brass well enough.
Hope this helps!