Answers

I would rather think your gutter needs a good sealant. If your soffit is sagging, you have some structural issues.Most likely nothing to do with the gutter. 

Hardi has about 30% cellulose fiber in it and the rest is cement. Hardi shrinks after it is installed so you must keep it tight to the the next board and not leave expansion gaps on either side. It definitely must have slip sheets under the joints. It must be caulked at the corner posts and openings. If you  cut it, it must be doped or painted as it is installed.

LP Smartside on the other hand expands after it is installed. Approximately 3/8 of an inch. horizontally for the length of each board. They have a gauge to place between each horizontal lap to allow for expansion. Here again I would use a slip sheet behnd the lap. Both prodcts are susceptible to moisture. So here again you must dope or paint any cuts as you install if you are using a predinished product. Both Hardi and LP require you to where masks to keep from breathing in the dust. Not good for your lungs.  

If you really want a good product with a superior finish, consider steel siding. Vinyl is just plastic and it breaks when hit by hail requiring it be replaced. Get a grill any where close to it when barbeciung and it melts.  Steel can dent in a major hailstorm, and it normally doesn't. but it still won't leak. Hardi board and LP will cost you more to install if prefinished than steel and  steel is virtually maintenance free. My house has steel on it since 1975. Most vinyls will last unitil the next big hailstorm and then you have to replace. The one drawback for steel is, it is harder to repair than vinyl. but can be done by a skilled craftsmen, without removing the whole wall. 

Windows that have a .28 u factor or. Slightly lower, if installed correctly sealing the perimeter with foam are generally adequate for most homes. They will have low -e and argon normally in the csvity between the insulated glass. Now if you want to go for triple glazed glass and low e and Krypton glass, you can get down to .17 u. Generally the additional cost and added weight do not pay for the added expense. You may do well to also add insulation in the attic and make sure you ventilate the soffit and fascia. Heat tends to travel up. Make sure you get a good installer that know s how to insulate and do a quality install. 

Did you do diligence to see who you were dealing with? Was the roofer licensed or required to be by the state or local unit of governement? ? Was a permit pulled and the job inspected by a building officia/ when completed? If not call your local building official and have them come out to look at the job? The roofer may be in violation of the bullding codes and be subject to disciplinary action by the state. I would check all these things before you proceed with a lawyer. Last if that contractor is not established, he won't have the money to fix it. But if he is insured you may be able to go after his insurance company. So aways ask to see a license, insurance and work comp certificate when dealing with any company and find out how long they have been in business prior to signing any contracts. 

Ideally, you will have no sweating if you have proper insulation and ventialtion. Generally moisture forms by ice freezing on the bottom side of the roof sheathng or the nails sticking through tje shething and when it melts you have water( sweating.) Not good. This can lead to wet insulation, mold and deterioration of the sheathing and or framing members. Get itchecked by a professional. 

You must have paid without inspecting the work. Building code requires a weather barrier under the siding and that it be taped at openings and all seams. Was there a permit obtained to do the work? If not, did your locality require one. If they did, report the contractor to the inspection department  as having not pulled a permit and gotten the right inspections. They may be able to help you and it oculd be a license violation in your state. Although most will refuse if a permit was not obtained and the proper inspections performed. Maybe they will go after the contractor and have them make it right. Have you checked your ceiling insulation? Heat rises and if the attic is not properly insulated(This is where 2/3 of the heat escapes from a house.  and  the soffits(overhang)need to be  vented. Heat will escape through the roof. How about your windows. Caulking around the j- channels where it meets the frame may stop some air and water infiltration.Do this with a clear caulking made for that purposed.  A lot of heat escapes through windows if they are not properly sealed. You may be able to put some plastic film over the windows and inside casings like 3-M. This may also help you with heat loss. There are foam inserts that you could put in the electrical outlets by removing the cover screw ,,pulling the cover,  putting them in place and then puttng the screw back in. Maybe the walls should have been insulated while you had the siding off. There are many things you havent't told us but these are ideas that may help your situation. It sounds like you do not have a foundation with blocks. But they are also a souce of heat loss. If you do, you could insulate the rim joists with a high density foam sheet and cut them to fit. and then use caulk or spray foam to seal around the edges. Hope this helps you.  

J.H. Fiber cement is composed of 70% concrete and 30% celluoose. Cellulose being ground up paper. Both water and cement wick moisture so if you use these products be sure to read and follow installation instructions. Leave no surface unpainted or it will draw moisture and deteriorate. Seen it many times. It is also dusty if you cut it with a saw. It is heavy, breaks easily if flexed. It is also heavy so it only comes in 12 foot lengths for the most part. It has the insulation value of cement which R-1 per inch. It is 5/16 thick so you get 5/16 or R-1 which is not much.

Vinyl is plastic but it is about half the price and any handyman can install if they follow directions so it can expand and contractl. Don't expect it top lasd forever. Hail storms beat it up pretty bad if the stones are big enough. One caution with it is do not place a grill near i, vinyl melts at 150 to 165 degerees depending on how thick it is. 

In my opinion steel is a little costiler but in most cases lasts for a long time.  If you figure it out, you will replace vinyl about every seven years. My stleel has been on my home in a hail region since 1975, No hail damage. You do the math.

If you are unsure check with the BBB. your state licensing board, and go look at several references where they have done instalations. Talk to the owners if you can and see how they were treated. These three things should tell you a lot about who you are dealing with. also have them give you the name of the brand of cement board and have them describe exactly what they propose to do. also look to see if one is prefinished and one is not. Prefinished are more expensive Good Luck.