Do effects like Shock and Burning still apply with Resolute Technique

The game will automatically assign new skills to any empty slots in your hotkey menu. However, if you click on any of the skill slots in the bottom right, you can assign/unassign whichever attack/skill you want.

Except in very rare cases (ignite and poison), damage modifiers will not double dip. The conversion is calculated first, then the increased and more multipliers. You can find several example conversions on the wiki page here: http://pathofexile.gamepedia.com/Damage_conversion

I'll copy the simplest one here for convenience.

The main benefit of converted damage is allowing bonuses to multiple damage types to stack. However, it does not allow one bonus to apply multiple times. This is because bonuses apply after conversion, but check for damage types that the damage had before conversion.

For example, start with 100 base Physical damage. Then convert 100% physical to Lightning to Cold to Fire:

"10% increased elemental damage"

will result in 110 fire damage. (If it was ever elemental damage, apply bonus) On the other hand,

"10% increased fire damage

10% increased lightning damage

10% increased cold damage"

will result in 130 fire damage. (For each line, if it was ever x damage type, apply bonus)

In the case of ignite and poison, increased/more fire and chaos damage respectively WILL effect the resulting ignite/poison damage twice, once on the initial hit, then again for the DoT calculation. This is what led to the recent chaos conversion nerf, and will likely lead to a nerf for burn builds once GGG notice how ridiculous the DPS from a grand spectrum burn build can get. (Especially if you manage to add in Emberwake, The Taming, and say Quillrain)

There is also one passive near the witch called Snowforged that adds two separate bonuses to fire and cold damage, and will therefore work with conversion twice.

In the case of your specific example, you would get 100% of your cold damage converted to fire, then an additional 10-29% extra fire from the cold to fire gem. So with a level 20 gem, 129% of your base cold would be dealt as fire, then any increased/more modifiers would be applied to that 129% total fire damage. If your cold to fire gem was 20 quality as well, it would add 10% increased fire, and 10% increased cold, both of which would apply to the final damage because they are separate bonuses.