In your original image you had an unused Service Area Layer tool which is a potential cause of error 735. If you still get 735 after removing that, it would indicate an unexposed, required parameter in the Make Layer tool isn't set (possibly impedence attribute based on your model screenshot). Just before the error number in the results it may give you more specific info.
Warning 030036 is just that, a warning that a layer name already exists and it will be overwritten. This will happen if you don't clear/delete stuff created by a run before running again (validate doesn't do this). It should not affect running the model unless you also get a write failure.
As for your lack of outputs, Network Analysis Layers (properly layer groups) are feature layers, not feature classes or shapefiles, which is what you load into them. They're constructs used to solve a problem, and it all takes place in memory/scratch. Results aren't permanent until you export them, much like a Create X,Y Event Layer. Your model currently selects lines, demand points, and facilities, but it doesn't do anything with that selection (there's also no reason to have those results as parameters). You need to add some sort of export tool, such as Copy Features after your select tools. It's there you would make a parameter of the output file to save permanent copy of the results.
Displaying the results is a different issue, as checking the Add to Display should add the results to a map. When I replicated your model, checking that box on your select outputs added them to the map with or without also adding the last version of the location allocation layer (you should try that to see if anything is being created) or anything already being in the map. If yours is not, that may indicate the solver isn't actually generating any results for some reason. It may also depend on how/where you are running the model from (ie, in Modelbuilder > Run Entire Model or just double-clicking it in ArcCatalog within toolbox).
I understand you want model the solution you have, however I want to offer an alternative method: run a 5 minute service area analysis to generate polys, then get counts of demand points per poly to find the three with the highest count. Location Allocation is more complex and has several variables (problem type, weighting) that I didn't fully explore in figuring out the modeling issue, but I did note they can change the results. You may already be familiar with them and have all the necessary details and decisions.
Also, it has you counting existing stations as possible locations. Unless they are considering abandoning existing stations and moving to 'optimal' locations, it seems the analysis would better find demand points not already served within five minutes and then pick proposed candidates that can reach the maximum number of those with minimal overlap of existing station service areas. But again, you obviously have more details and information about the problem and it may be a different one than I'm interpreting.
