yes , you can play on the net , i recomend you go through and reinstall your steam/hl/esf & the patches , or tweak your net connection/browser settings& or system settings , if all else fails , time to reformat the dam sun of a gun , and if that doesn't work , the dumpster with it (i could imagine your annoyence at your pc , "you stupiud machine , to hell with you" , lo , only playying )
then, you can download hl directly through steam or import your local (old) hl installation (right click on Half-Life in the "My Games" List after you registred it!).
then you can either reinstall or copy&paste your old mods to your Steam HL diretory i.e. c: \steam\stamapps\youraccount\half-life\