Yes RSM you can upgrade to those which you mentioned also as your Punto is designed by Fiat themselves , and so there wont be any handling or suspension problems. Only marginal decrease in FE and also slight speedo error.
When you upsize tyres, your ride quality goes down/up. That is, as you go higher on the rim size & tyre width, the sidewall thickness reduces. Now the majority of the impact on bad roads is not absorbed by the higher sidewall on your previous tyre but the brunt is borne by your suspension after the upgrade. So yes, it does affect suspension.
How it affects handling? Improvement in handling is more to do with the *tyre compound* than size. For eg a soft compound tyre in 185/70/14 can give better grip than a 205/60/14 with a harder compound. Size and compound of tyre help not only in handling, but more importantly in reducing braking distances. This is one aspect we Indians usually tend to overlook while upgrading tyres.
So yes, tyre upsize does have an impact on suspension. And a wrong upsize can cause issues.
For eg, using a 225 width tyre on a 1.2L 75bhp car will make it very slow. You will have stock cars overtaking you.
Putting a 16'' rim with a sidewall of less than 50 profile will make the ride very harsh. This in turn will affect your suspension and other related components too. This can even cause more rattles from your cars interiors too.
To know the difference drive a car with low profile tyres over bad roads and see.
Does rim size (Punto has 5.5j) matters for deciding the width of the tyre?
Definitely. The rim width decides the max width of the tyre you can use. If the rim size is inadequate, you will have your sidewalls bulging on both sides of the rim. This will not only result in lesser sidewall strength, it also puts undue strain on the sidewalls. The other disadvantage being, you get lesser contact patch available.
For the Punto, the bigger problem is the PCD is 98. Which makes it extremely tough to get good alloys if you want to go for 195 width or upgrade to 15s.
Suggested upgrades would be:
185/65/14
195/65-14
195/55/15
I haven't checked the error in OD on these, but as long as its under 3% you are good to go. Anything below 55 profile or higher than 15" rims is suicidal on Indian roads.