I am finishing my basement which will include a bar. I would like to place a small 2 burner built-in cooktop rated at 20 amp (110 v). Do I need to place this on a dedicated circuit? Can I place this on my soon to be shop circuit which will be 20 amp breaker 12-2 wire with about 4 outlets only? ( I am sure the shop outlets will not be in use when the cooktop is)

I'd recommend a dedicated circuit for the cooktop since it's rated at 20 amps. Anything else on the circuit will cause and overload. I know you said the other outlets would not be used while the cooktop is in use but this is not a 100% guarantee down the road.

OK, this may cause a problem. I have a 100 amp main with 20 slot GE box, all full. I have read I can have max 48 breakers? So I am planning on already pulling 2 of the 20 amp and replacing them with 4 GE 1/2" breakers and then doing the same with 2 of the 15 amp breakers. so will pulling another 20 amp for cooktop circuit be a problem? this will leave me 1 open 1/2" slot for total of 26 breakers.

Are you saying that you will be using one of these 1/2 sized breakers for the cooktop or are you doing that switch out for some other reason?

This somewhat depends on how you count. There isn't a breaker limit, but rather a pole limit. e.g. a 240 volt breaker has two poles, so it counts as 2. That number should not be exceeded.

You can add a small sub-panel, say 60A if you needed too.

I would need to free up a slot somehow, but I would probably try to find a circuit that can be placed on a 1/2" rather than a dedicated one

Kiss, would the subpanel be preferred over the 10 1/2" breakers?

The half sized breakers are safe, UL approved and cheaper/easier than a sub.

The panel will only accept skinny breaker if it is designed to. You can't simply put them in anywhere.
If it is a 100A panel it is most likely that you cannot out hem in at all, and if you can probably only in the bottom 4 spaces.

The 42 pole limit is over all. Not in every panel.