My answer is yes. As abstracted in the question, the peptide will have a 3D structure (molecular spacial conformation) which contains an alpha helix (secondary structure).

You can try to do it. If the precise sequence of the peptide AA is known, this would be helpful to define its tertiary structure (3D).

good luck

1 - it does not matter what structure your peptide has

2 - definition: IBs are the target protein in not native form, it is not clear why you need to make it bounded to the insoluble partner, actually for lab scale purification people always try to avoid the IBs formation by getting the soluble form which does not require refolding step, if you add the insoluble partner then it does not mean that your target peptide will be in the IB-form (not native form) - it can be in the soluble form (native form) attached to the insoluble partner and in this case you need the restriction step but not the refolding one, additional to that even the native form of your peptide without the insoluble partner can be insoluble because just solubility of your peptide is very low and it might look like "IBs" or can be soluble/insoluble outside of the cells

3 - peptide definitely can be refolded because it does not need post - gluconasation that mostly needed for large mammalian proteins

I agree with Anatoliy here. In my experience it is best to get the peptide/protein to express in a soluble form. Sometimes that means using an alternate expression system, editing your plasmid, moving an affinity tag, or experimenting with various buffer systems for lysis and elution. Refolding should be a last resort, but in the case of your tiny peptide a simple GF run with a urea gradient would probably suffice. I would not, however, specifically tag my sequence to an insoluble tag and proceed through IB. 

An alternate route is to get your peptide synthesized. I went to:http://biomatik.com/AppAction/OLPepQuoteSimulator.aspx, entered a fictitious peptide sequence of 31 standard proteinogenic amino acids, and instantly received a quote for $507 for 50 mg of protein (I have no affiliation with the aforementioned company). Depending on your current funding situation this may be a viable alternative to fussing with the peptide expression. 

Thank you for your precious opinions. I want to produce this small peptide by E. coli/IB system to avoid from protease/peptidase attack, since peptides are more sensitive to protease/peptidase in a soluble circumstance. I think that microbial expression system is a better way to make peptide than chemical synthesis in terms of scale up prodution. What I am concerned is whether the peptide purified from IB can get the right structure with its native bioactivity?

As I told before, your peptide from E coli/IB will have a correct conformation if it is expressed correctly and has the same AA sequence as that of the natively produced form. The experiment will give you the answer. So it is worthy to try it.

My encouragement.

Regardless of whether you obtain your peptide from IB or in soluble form, you will still need to have an activity assay ready to convince people that your peptide is biologically active.

I also think that Carter's suggestion about buying synthesised peptides are quite relevant, considering that it could save you a lot of time and maybe even money down the stretch. On the other hand, if you have plans to introduce mutations in your peptide, then the microbial expression would be the way to go.

try PEPstrMOD

PEPstrMOD: structure prediction of peptides containing natural, non-natural and modified residues (http://osddlinux.osdd.net/raghava/pepstrmod/)

article link: http://www.biologydirect.com/content/10/1/73