I've done it for one check, not all three, but you will see how it works.

You're on the right track, but you need to seperate out your processes:

1) Get all accounts from all leads matching your criteria ( "f(l.LeadSource == 'Sign Up' && l.Was_Invited__c=='TRUE' && !l.isConverted) " )

2) Create a map of all of those accounts by name so you can do a quick map.get(companyName) to check if it's null

3) Only then create your lead

You're trying to do all this at once

So here if my sample - not that I didn't test it, jsut wrote it out:

trigger myLeadConvertTrigger on Lead(after insert, after update) {
	//Create an empty list of Leads to convert
    Database.LeadConvert[] leadsToConvert = new Database.LeadConvert[]{};
    //Find a Lead Status for Leads converted - note that this will take 1 value and only the first one that it finds
    LeadStatus convertStatus = [SELECT Id, MasterLabel FROM LeadStatus WHERE IsConverted=true LIMIT 1];
    
	//Create a map of lead ids, to account names
	set<string> accountNames = new set<string>();

	list<Lead> leads = new list<lead>();
    //For Lead being updated or inserted
    for(Lead l : Trigger.new) {
        
        // if statement logic: Lead is Lead Source 'Sign Up' and was Invited and isn't converted yet
        if(l.LeadSource == 'Sign Up' && l.Was_Invited__c=='TRUE' && !l.isConverted) {
			leads.add(lead);

			//Add the various identifiers to their maps to check against later
			accountNames.add(l.Company);
			//Create two more sets, and add the criteria you are checking against. It may be prudent to check for null values first! We know company is a required fields, but you will get strange results if you add 'null' to the sets!
		}
	}
	
	//If we have leads to convert:
	if(!leads.isEmpty()) {
		//Query for the various lists we need - Note that calling .values() on a map returns a list of all values in the whole map
		Account[] accounts = [SELECT Id, Name FROM Account WHERE Name IN :accountNames];
		
		//And for each of those lists, create a corresponding map to check against
		map<string, account> accountNames_map = new map<string,account>();
		for(account a : accounts) {
			accountNames_map.put(a.name,a);
		}
		
		
		//And for each lead, find if it has a match
		for(Lead l : leads) {
			//Create a new Lead Convert Record
            Database.LeadConvert lc = new database.LeadConvert();
            lc.setLeadId(l.id);
            lc.setConvertedStatus(convertStatus.MasterLabel); //set it to the status we retrieved
            lc.setDoNotCreateOpportunity(True); //do not create an opp
			
			if(accountNames_map.get(l.company) != null) { //Hey we have that company already!!!
				lc.setAccountId(accountNames_map.get(l.company));
			}
		}
	}

	//if the leadsToConvert list is not empty
    if(!leadsToConvert.isEmpty()) {
        //convert those Leads
        Database.convertLead(leadsToConvert );
    }
}

So you really just need two more maps for each of those other queries