Hi I want to switch up my cog/chain ring because where I live now is all flat. Right now I run a 44/19 and I dont know whether to get a smaller cog or bigger chain ring. If I keep chainring I was looking at possibly 13 tooth would that be good? Also would this work?

If I would change the chain ring I was looking at possibly 50 teeth. Would either of these work?

I know how a cog goes on so I would just take it to my bike shop and last time they put my cog and lockring on for free since I dont have these tools, but how is a chain ring changed and how much do bike shops charge regularly?

I also want to remove my rear brake and change the placement of my lever for my front brake. I have pursuit handlebars that are setup like this currently and I want to change the brake lever to something like this where the lever is on the horizontal part. Any advice on how to do this or guides anybody could point me towards? Also I assume I would have to buy a different brake lever holder thingy (IDK what they're called) so what would I need for this? Thanks guys

EDIT: After looking at the numbers on a fixed gear calculator I realize keeping a 19T cog and just 50T chainring won't make much difference, so I would either go 13T and keep my 44T or get both new ones. Any suggestions on which would be better?